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Theorem txhaus 23590

Description: The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)

**Assertion**
Ref	Expression
txhaus	⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×_t 𝑆) ∈ Haus)

Proof of Theorem txhaus Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		haustop 23274	. . 3 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Top)
2		haustop 23274	. . 3 ⊢ (𝑆 ∈ Haus → 𝑆 ∈ Top)
3		txtop 23512	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×_t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×_t 𝑆) ∈ Top)
5		eqid 2736	. . . . . . . 8 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2736	. . . . . . . 8 ⊢ ∪ 𝑆 = ∪ 𝑆
7	5, 6	txuni 23535	. . . . . . 7 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
8	1, 2, 7	syl2an 596	. . . . . 6 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×_t 𝑆))
9	8	eleq2d 2821	. . . . 5 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ↔ 𝑥 ∈ ∪ (𝑅 ×_t 𝑆)))
10	8	eleq2d 2821	. . . . 5 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑦 ∈ (∪ 𝑅 × ∪ 𝑆) ↔ 𝑦 ∈ ∪ (𝑅 ×_t 𝑆)))
11	9, 10	anbi12d 632	. . . 4 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ((𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆)) ↔ (𝑥 ∈ ∪ (𝑅 ×_t 𝑆) ∧ 𝑦 ∈ ∪ (𝑅 ×_t 𝑆))))
12		neorian 3028	. . . . . . 7 ⊢ (((1^st ‘𝑥) ≠ (1^st ‘𝑦) ∨ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ↔ ¬ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
13		xpopth 8034	. . . . . . . . 9 ⊢ ((𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆)) → (((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ↔ 𝑥 = 𝑦))
14	13	adantl 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ↔ 𝑥 = 𝑦))
15	14	necon3bbid 2970	. . . . . . 7 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (¬ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ↔ 𝑥 ≠ 𝑦))
16	12, 15	bitrid 283	. . . . . 6 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (((1^st ‘𝑥) ≠ (1^st ‘𝑦) ∨ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ↔ 𝑥 ≠ 𝑦))
17		simplll 774	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) → 𝑅 ∈ Haus)
18		xp1st 8025	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) → (1^st ‘𝑥) ∈ ∪ 𝑅)
19	18	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (1^st ‘𝑥) ∈ ∪ 𝑅)
20	19	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) → (1^st ‘𝑥) ∈ ∪ 𝑅)
21		xp1st 8025	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (∪ 𝑅 × ∪ 𝑆) → (1^st ‘𝑦) ∈ ∪ 𝑅)
22	21	ad2antll 729	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (1^st ‘𝑦) ∈ ∪ 𝑅)
23	22	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) → (1^st ‘𝑦) ∈ ∪ 𝑅)
24		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) → (1^st ‘𝑥) ≠ (1^st ‘𝑦))
25	5	hausnei 23271	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Haus ∧ ((1^st ‘𝑥) ∈ ∪ 𝑅 ∧ (1^st ‘𝑦) ∈ ∪ 𝑅 ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦))) → ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑅 ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
26	17, 20, 23, 24, 25	syl13anc 1374	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) → ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑅 ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
27	1	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → 𝑅 ∈ Top)
28	27	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑅 ∈ Top)
29	2	ad4antlr 733	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑆 ∈ Top)
30		simprll 778	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑢 ∈ 𝑅)
31	6	topopn 22849	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
32	29, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ∪ 𝑆 ∈ 𝑆)
33		txopn 23545	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑢 ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆)) → (𝑢 × ∪ 𝑆) ∈ (𝑅 ×_t 𝑆))
34	28, 29, 30, 32, 33	syl22anc 838	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝑢 × ∪ 𝑆) ∈ (𝑅 ×_t 𝑆))
35		simprlr 779	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑣 ∈ 𝑅)
36		txopn 23545	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑣 ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆)) → (𝑣 × ∪ 𝑆) ∈ (𝑅 ×_t 𝑆))
37	28, 29, 35, 32, 36	syl22anc 838	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝑣 × ∪ 𝑆) ∈ (𝑅 ×_t 𝑆))
38		1st2nd2 8032	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
39	38	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
40	39	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
41		simprr1 1222	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (1^st ‘𝑥) ∈ 𝑢)
42		xp2nd 8026	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) → (2^nd ‘𝑥) ∈ ∪ 𝑆)
43	42	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (2^nd ‘𝑥) ∈ ∪ 𝑆)
44	43	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (2^nd ‘𝑥) ∈ ∪ 𝑆)
45	41, 44	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((1^st ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑥) ∈ ∪ 𝑆))
46		elxp6 8027	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑢 × ∪ 𝑆) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑥) ∈ ∪ 𝑆)))
47	40, 45, 46	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑥 ∈ (𝑢 × ∪ 𝑆))
48		1st2nd2 8032	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (∪ 𝑅 × ∪ 𝑆) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
49	48	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
50	49	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
51		simprr2 1223	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (1^st ‘𝑦) ∈ 𝑣)
52		xp2nd 8026	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (∪ 𝑅 × ∪ 𝑆) → (2^nd ‘𝑦) ∈ ∪ 𝑆)
53	52	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (2^nd ‘𝑦) ∈ ∪ 𝑆)
54	53	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (2^nd ‘𝑦) ∈ ∪ 𝑆)
55	51, 54	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((1^st ‘𝑦) ∈ 𝑣 ∧ (2^nd ‘𝑦) ∈ ∪ 𝑆))
56		elxp6 8027	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑣 × ∪ 𝑆) ↔ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ 𝑣 ∧ (2^nd ‘𝑦) ∈ ∪ 𝑆)))
57	50, 55, 56	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑦 ∈ (𝑣 × ∪ 𝑆))
58		simprr3 1224	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝑢 ∩ 𝑣) = ∅)
59	58	xpeq1d 5688	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((𝑢 ∩ 𝑣) × ∪ 𝑆) = (∅ × ∪ 𝑆))
60		xpindir 5819	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑣) × ∪ 𝑆) = ((𝑢 × ∪ 𝑆) ∩ (𝑣 × ∪ 𝑆))
61		0xp 5758	. . . . . . . . . . . . 13 ⊢ (∅ × ∪ 𝑆) = ∅
62	59, 60, 61	3eqtr3g 2794	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((𝑢 × ∪ 𝑆) ∩ (𝑣 × ∪ 𝑆)) = ∅)
63		eleq2 2824	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑢 × ∪ 𝑆) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑢 × ∪ 𝑆)))
64		ineq1 4193	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑢 × ∪ 𝑆) → (𝑧 ∩ 𝑤) = ((𝑢 × ∪ 𝑆) ∩ 𝑤))
65	64	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑢 × ∪ 𝑆) → ((𝑧 ∩ 𝑤) = ∅ ↔ ((𝑢 × ∪ 𝑆) ∩ 𝑤) = ∅))
66	63, 65	3anbi13d 1440	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑢 × ∪ 𝑆) → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) ↔ (𝑥 ∈ (𝑢 × ∪ 𝑆) ∧ 𝑦 ∈ 𝑤 ∧ ((𝑢 × ∪ 𝑆) ∩ 𝑤) = ∅)))
67		eleq2 2824	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑣 × ∪ 𝑆) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (𝑣 × ∪ 𝑆)))
68		ineq2 4194	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑣 × ∪ 𝑆) → ((𝑢 × ∪ 𝑆) ∩ 𝑤) = ((𝑢 × ∪ 𝑆) ∩ (𝑣 × ∪ 𝑆)))
69	68	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑣 × ∪ 𝑆) → (((𝑢 × ∪ 𝑆) ∩ 𝑤) = ∅ ↔ ((𝑢 × ∪ 𝑆) ∩ (𝑣 × ∪ 𝑆)) = ∅))
70	67, 69	3anbi23d 1441	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑣 × ∪ 𝑆) → ((𝑥 ∈ (𝑢 × ∪ 𝑆) ∧ 𝑦 ∈ 𝑤 ∧ ((𝑢 × ∪ 𝑆) ∩ 𝑤) = ∅) ↔ (𝑥 ∈ (𝑢 × ∪ 𝑆) ∧ 𝑦 ∈ (𝑣 × ∪ 𝑆) ∧ ((𝑢 × ∪ 𝑆) ∩ (𝑣 × ∪ 𝑆)) = ∅)))
71	66, 70	rspc2ev 3619	. . . . . . . . . . . 12 ⊢ (((𝑢 × ∪ 𝑆) ∈ (𝑅 ×_t 𝑆) ∧ (𝑣 × ∪ 𝑆) ∈ (𝑅 ×_t 𝑆) ∧ (𝑥 ∈ (𝑢 × ∪ 𝑆) ∧ 𝑦 ∈ (𝑣 × ∪ 𝑆) ∧ ((𝑢 × ∪ 𝑆) ∩ (𝑣 × ∪ 𝑆)) = ∅)) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
72	34, 37, 47, 57, 62, 71	syl113anc 1384	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ ((𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅) ∧ ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
73	72	expr 456	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) ∧ (𝑢 ∈ 𝑅 ∧ 𝑣 ∈ 𝑅)) → (((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
74	73	rexlimdvva 3202	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) → (∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑅 ((1^st ‘𝑥) ∈ 𝑢 ∧ (1^st ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
75	26, 74	mpd 15	. . . . . . . 8 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (1^st ‘𝑥) ≠ (1^st ‘𝑦)) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
76		simpllr 775	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → 𝑆 ∈ Haus)
77	43	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → (2^nd ‘𝑥) ∈ ∪ 𝑆)
78	53	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → (2^nd ‘𝑦) ∈ ∪ 𝑆)
79		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → (2^nd ‘𝑥) ≠ (2^nd ‘𝑦))
80	6	hausnei 23271	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Haus ∧ ((2^nd ‘𝑥) ∈ ∪ 𝑆 ∧ (2^nd ‘𝑦) ∈ ∪ 𝑆 ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦))) → ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ 𝑆 ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
81	76, 77, 78, 79, 80	syl13anc 1374	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ 𝑆 ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))
82	27	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑅 ∈ Top)
83	2	ad4antlr 733	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑆 ∈ Top)
84	5	topopn 22849	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
85	82, 84	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ∪ 𝑅 ∈ 𝑅)
86		simprll 778	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑢 ∈ 𝑆)
87		txopn 23545	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (∪ 𝑅 ∈ 𝑅 ∧ 𝑢 ∈ 𝑆)) → (∪ 𝑅 × 𝑢) ∈ (𝑅 ×_t 𝑆))
88	82, 83, 85, 86, 87	syl22anc 838	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (∪ 𝑅 × 𝑢) ∈ (𝑅 ×_t 𝑆))
89		simprlr 779	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑣 ∈ 𝑆)
90		txopn 23545	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (∪ 𝑅 ∈ 𝑅 ∧ 𝑣 ∈ 𝑆)) → (∪ 𝑅 × 𝑣) ∈ (𝑅 ×_t 𝑆))
91	82, 83, 85, 89, 90	syl22anc 838	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (∪ 𝑅 × 𝑣) ∈ (𝑅 ×_t 𝑆))
92	39	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
93	19	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (1^st ‘𝑥) ∈ ∪ 𝑅)
94		simprr1 1222	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (2^nd ‘𝑥) ∈ 𝑢)
95	93, 94	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((1^st ‘𝑥) ∈ ∪ 𝑅 ∧ (2^nd ‘𝑥) ∈ 𝑢))
96		elxp6 8027	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (∪ 𝑅 × 𝑢) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ∪ 𝑅 ∧ (2^nd ‘𝑥) ∈ 𝑢)))
97	92, 95, 96	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑥 ∈ (∪ 𝑅 × 𝑢))
98	49	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
99	22	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (1^st ‘𝑦) ∈ ∪ 𝑅)
100		simprr2 1223	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (2^nd ‘𝑦) ∈ 𝑣)
101	99, 100	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((1^st ‘𝑦) ∈ ∪ 𝑅 ∧ (2^nd ‘𝑦) ∈ 𝑣))
102		elxp6 8027	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (∪ 𝑅 × 𝑣) ↔ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ∪ 𝑅 ∧ (2^nd ‘𝑦) ∈ 𝑣)))
103	98, 101, 102	sylanbrc 583	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → 𝑦 ∈ (∪ 𝑅 × 𝑣))
104		simprr3 1224	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (𝑢 ∩ 𝑣) = ∅)
105	104	xpeq2d 5689	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → (∪ 𝑅 × (𝑢 ∩ 𝑣)) = (∪ 𝑅 × ∅))
106		xpindi 5818	. . . . . . . . . . . . 13 ⊢ (∪ 𝑅 × (𝑢 ∩ 𝑣)) = ((∪ 𝑅 × 𝑢) ∩ (∪ 𝑅 × 𝑣))
107		xp0 6152	. . . . . . . . . . . . 13 ⊢ (∪ 𝑅 × ∅) = ∅
108	105, 106, 107	3eqtr3g 2794	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ((∪ 𝑅 × 𝑢) ∩ (∪ 𝑅 × 𝑣)) = ∅)
109		eleq2 2824	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (∪ 𝑅 × 𝑢) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (∪ 𝑅 × 𝑢)))
110		ineq1 4193	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (∪ 𝑅 × 𝑢) → (𝑧 ∩ 𝑤) = ((∪ 𝑅 × 𝑢) ∩ 𝑤))
111	110	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (∪ 𝑅 × 𝑢) → ((𝑧 ∩ 𝑤) = ∅ ↔ ((∪ 𝑅 × 𝑢) ∩ 𝑤) = ∅))
112	109, 111	3anbi13d 1440	. . . . . . . . . . . . 13 ⊢ (𝑧 = (∪ 𝑅 × 𝑢) → ((𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) ↔ (𝑥 ∈ (∪ 𝑅 × 𝑢) ∧ 𝑦 ∈ 𝑤 ∧ ((∪ 𝑅 × 𝑢) ∩ 𝑤) = ∅)))
113		eleq2 2824	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (∪ 𝑅 × 𝑣) → (𝑦 ∈ 𝑤 ↔ 𝑦 ∈ (∪ 𝑅 × 𝑣)))
114		ineq2 4194	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (∪ 𝑅 × 𝑣) → ((∪ 𝑅 × 𝑢) ∩ 𝑤) = ((∪ 𝑅 × 𝑢) ∩ (∪ 𝑅 × 𝑣)))
115	114	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (∪ 𝑅 × 𝑣) → (((∪ 𝑅 × 𝑢) ∩ 𝑤) = ∅ ↔ ((∪ 𝑅 × 𝑢) ∩ (∪ 𝑅 × 𝑣)) = ∅))
116	113, 115	3anbi23d 1441	. . . . . . . . . . . . 13 ⊢ (𝑤 = (∪ 𝑅 × 𝑣) → ((𝑥 ∈ (∪ 𝑅 × 𝑢) ∧ 𝑦 ∈ 𝑤 ∧ ((∪ 𝑅 × 𝑢) ∩ 𝑤) = ∅) ↔ (𝑥 ∈ (∪ 𝑅 × 𝑢) ∧ 𝑦 ∈ (∪ 𝑅 × 𝑣) ∧ ((∪ 𝑅 × 𝑢) ∩ (∪ 𝑅 × 𝑣)) = ∅)))
117	112, 116	rspc2ev 3619	. . . . . . . . . . . 12 ⊢ (((∪ 𝑅 × 𝑢) ∈ (𝑅 ×_t 𝑆) ∧ (∪ 𝑅 × 𝑣) ∈ (𝑅 ×_t 𝑆) ∧ (𝑥 ∈ (∪ 𝑅 × 𝑢) ∧ 𝑦 ∈ (∪ 𝑅 × 𝑣) ∧ ((∪ 𝑅 × 𝑢) ∩ (∪ 𝑅 × 𝑣)) = ∅)) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
118	88, 91, 97, 103, 108, 117	syl113anc 1384	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅))) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
119	118	expr 456	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) ∧ (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆)) → (((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
120	119	rexlimdvva 3202	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → (∃𝑢 ∈ 𝑆 ∃𝑣 ∈ 𝑆 ((2^nd ‘𝑥) ∈ 𝑢 ∧ (2^nd ‘𝑦) ∈ 𝑣 ∧ (𝑢 ∩ 𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
121	81, 120	mpd 15	. . . . . . . 8 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
122	75, 121	jaodan 959	. . . . . . 7 ⊢ ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) ∧ ((1^st ‘𝑥) ≠ (1^st ‘𝑦) ∨ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦))) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
123	122	ex 412	. . . . . 6 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (((1^st ‘𝑥) ≠ (1^st ‘𝑦) ∨ (2^nd ‘𝑥) ≠ (2^nd ‘𝑦)) → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
124	16, 123	sylbird 260	. . . . 5 ⊢ (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆))) → (𝑥 ≠ 𝑦 → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
125	124	ex 412	. . . 4 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ((𝑥 ∈ (∪ 𝑅 × ∪ 𝑆) ∧ 𝑦 ∈ (∪ 𝑅 × ∪ 𝑆)) → (𝑥 ≠ 𝑦 → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))))
126	11, 125	sylbird 260	. . 3 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ((𝑥 ∈ ∪ (𝑅 ×_t 𝑆) ∧ 𝑦 ∈ ∪ (𝑅 ×_t 𝑆)) → (𝑥 ≠ 𝑦 → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))))
127	126	ralrimivv 3186	. 2 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ∀𝑥 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑦 ∈ ∪ (𝑅 ×_t 𝑆)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))
128		eqid 2736	. . 3 ⊢ ∪ (𝑅 ×_t 𝑆) = ∪ (𝑅 ×_t 𝑆)
129	128	ishaus 23265	. 2 ⊢ ((𝑅 ×_t 𝑆) ∈ Haus ↔ ((𝑅 ×_t 𝑆) ∈ Top ∧ ∀𝑥 ∈ ∪ (𝑅 ×_t 𝑆)∀𝑦 ∈ ∪ (𝑅 ×_t 𝑆)(𝑥 ≠ 𝑦 → ∃𝑧 ∈ (𝑅 ×_t 𝑆)∃𝑤 ∈ (𝑅 ×_t 𝑆)(𝑥 ∈ 𝑧 ∧ 𝑦 ∈ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))))
130	4, 127, 129	sylanbrc 583	1 ⊢ ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×_t 𝑆) ∈ Haus)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 ∩ cin 3930 ∅c0 4313 ⟨cop 4612 ∪ cuni 4888 × cxp 5657 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 Topctop 22836 Hauscha 23251 ×_t ctx 23503

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-topgen 17462 df-top 22837 df-topon 22854 df-bases 22889 df-haus 23258 df-tx 23505

This theorem is referenced by: (None)

W3C validator