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Theorem txhaus 21498
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.
StepHypRef Expression
1 haustop 21183 . . 3 (𝑅 ∈ Haus → 𝑅 ∈ Top)
2 haustop 21183 . . 3 (𝑆 ∈ Haus → 𝑆 ∈ Top)
3 txtop 21420 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 493 . 2 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2651 . . . . . . . 8 𝑅 = 𝑅
6 eqid 2651 . . . . . . . 8 𝑆 = 𝑆
75, 6txuni 21443 . . . . . . 7 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
81, 2, 7syl2an 493 . . . . . 6 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
98eleq2d 2716 . . . . 5 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑥 ∈ ( 𝑅 × 𝑆) ↔ 𝑥 (𝑅 ×t 𝑆)))
108eleq2d 2716 . . . . 5 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑦 ∈ ( 𝑅 × 𝑆) ↔ 𝑦 (𝑅 ×t 𝑆)))
119, 10anbi12d 747 . . . 4 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ((𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆)) ↔ (𝑥 (𝑅 ×t 𝑆) ∧ 𝑦 (𝑅 ×t 𝑆))))
12 neorian 2917 . . . . . . 7 (((1st𝑥) ≠ (1st𝑦) ∨ (2nd𝑥) ≠ (2nd𝑦)) ↔ ¬ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
13 xpopth 7251 . . . . . . . . 9 ((𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆)) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥 = 𝑦))
1413adantl 481 . . . . . . . 8 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥 = 𝑦))
1514necon3bbid 2860 . . . . . . 7 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (¬ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)) ↔ 𝑥𝑦))
1612, 15syl5bb 272 . . . . . 6 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (((1st𝑥) ≠ (1st𝑦) ∨ (2nd𝑥) ≠ (2nd𝑦)) ↔ 𝑥𝑦))
17 simplll 813 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) → 𝑅 ∈ Haus)
18 xp1st 7242 . . . . . . . . . . . 12 (𝑥 ∈ ( 𝑅 × 𝑆) → (1st𝑥) ∈ 𝑅)
1918ad2antrl 764 . . . . . . . . . . 11 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (1st𝑥) ∈ 𝑅)
2019adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ∈ 𝑅)
21 xp1st 7242 . . . . . . . . . . . 12 (𝑦 ∈ ( 𝑅 × 𝑆) → (1st𝑦) ∈ 𝑅)
2221ad2antll 765 . . . . . . . . . . 11 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (1st𝑦) ∈ 𝑅)
2322adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑦) ∈ 𝑅)
24 simpr 476 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ≠ (1st𝑦))
255hausnei 21180 . . . . . . . . . 10 ((𝑅 ∈ Haus ∧ ((1st𝑥) ∈ 𝑅 ∧ (1st𝑦) ∈ 𝑅 ∧ (1st𝑥) ≠ (1st𝑦))) → ∃𝑢𝑅𝑣𝑅 ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))
2617, 20, 23, 24, 25syl13anc 1368 . . . . . . . . 9 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) → ∃𝑢𝑅𝑣𝑅 ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))
271ad2antrr 762 . . . . . . . . . . . . . 14 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → 𝑅 ∈ Top)
2827ad2antrr 762 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑅 ∈ Top)
292ad4antlr 771 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑆 ∈ Top)
30 simprll 819 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑢𝑅)
316topopn 20759 . . . . . . . . . . . . . 14 (𝑆 ∈ Top → 𝑆𝑆)
3229, 31syl 17 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑆𝑆)
33 txopn 21453 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑢𝑅 𝑆𝑆)) → (𝑢 × 𝑆) ∈ (𝑅 ×t 𝑆))
3428, 29, 30, 32, 33syl22anc 1367 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (𝑢 × 𝑆) ∈ (𝑅 ×t 𝑆))
35 simprlr 820 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑣𝑅)
36 txopn 21453 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑣𝑅 𝑆𝑆)) → (𝑣 × 𝑆) ∈ (𝑅 ×t 𝑆))
3728, 29, 35, 32, 36syl22anc 1367 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (𝑣 × 𝑆) ∈ (𝑅 ×t 𝑆))
38 1st2nd2 7249 . . . . . . . . . . . . . . 15 (𝑥 ∈ ( 𝑅 × 𝑆) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
3938ad2antrl 764 . . . . . . . . . . . . . 14 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
4039ad2antrr 762 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
41 simprr1 1129 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (1st𝑥) ∈ 𝑢)
42 xp2nd 7243 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ( 𝑅 × 𝑆) → (2nd𝑥) ∈ 𝑆)
4342ad2antrl 764 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (2nd𝑥) ∈ 𝑆)
4443ad2antrr 762 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (2nd𝑥) ∈ 𝑆)
4541, 44jca 553 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ((1st𝑥) ∈ 𝑢 ∧ (2nd𝑥) ∈ 𝑆))
46 elxp6 7244 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑢 × 𝑆) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ 𝑢 ∧ (2nd𝑥) ∈ 𝑆)))
4740, 45, 46sylanbrc 699 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑥 ∈ (𝑢 × 𝑆))
48 1st2nd2 7249 . . . . . . . . . . . . . . 15 (𝑦 ∈ ( 𝑅 × 𝑆) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4948ad2antll 765 . . . . . . . . . . . . . 14 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
5049ad2antrr 762 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
51 simprr2 1130 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (1st𝑦) ∈ 𝑣)
52 xp2nd 7243 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ( 𝑅 × 𝑆) → (2nd𝑦) ∈ 𝑆)
5352ad2antll 765 . . . . . . . . . . . . . . 15 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (2nd𝑦) ∈ 𝑆)
5453ad2antrr 762 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (2nd𝑦) ∈ 𝑆)
5551, 54jca 553 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ((1st𝑦) ∈ 𝑣 ∧ (2nd𝑦) ∈ 𝑆))
56 elxp6 7244 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑣 × 𝑆) ↔ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ 𝑣 ∧ (2nd𝑦) ∈ 𝑆)))
5750, 55, 56sylanbrc 699 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑦 ∈ (𝑣 × 𝑆))
58 simprr3 1131 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (𝑢𝑣) = ∅)
5958xpeq1d 5172 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ((𝑢𝑣) × 𝑆) = (∅ × 𝑆))
60 xpindir 5289 . . . . . . . . . . . . 13 ((𝑢𝑣) × 𝑆) = ((𝑢 × 𝑆) ∩ (𝑣 × 𝑆))
61 0xp 5233 . . . . . . . . . . . . 13 (∅ × 𝑆) = ∅
6259, 60, 613eqtr3g 2708 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ((𝑢 × 𝑆) ∩ (𝑣 × 𝑆)) = ∅)
63 eleq2 2719 . . . . . . . . . . . . . 14 (𝑧 = (𝑢 × 𝑆) → (𝑥𝑧𝑥 ∈ (𝑢 × 𝑆)))
64 ineq1 3840 . . . . . . . . . . . . . . 15 (𝑧 = (𝑢 × 𝑆) → (𝑧𝑤) = ((𝑢 × 𝑆) ∩ 𝑤))
6564eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑧 = (𝑢 × 𝑆) → ((𝑧𝑤) = ∅ ↔ ((𝑢 × 𝑆) ∩ 𝑤) = ∅))
6663, 653anbi13d 1441 . . . . . . . . . . . . 13 (𝑧 = (𝑢 × 𝑆) → ((𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅) ↔ (𝑥 ∈ (𝑢 × 𝑆) ∧ 𝑦𝑤 ∧ ((𝑢 × 𝑆) ∩ 𝑤) = ∅)))
67 eleq2 2719 . . . . . . . . . . . . . 14 (𝑤 = (𝑣 × 𝑆) → (𝑦𝑤𝑦 ∈ (𝑣 × 𝑆)))
68 ineq2 3841 . . . . . . . . . . . . . . 15 (𝑤 = (𝑣 × 𝑆) → ((𝑢 × 𝑆) ∩ 𝑤) = ((𝑢 × 𝑆) ∩ (𝑣 × 𝑆)))
6968eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑤 = (𝑣 × 𝑆) → (((𝑢 × 𝑆) ∩ 𝑤) = ∅ ↔ ((𝑢 × 𝑆) ∩ (𝑣 × 𝑆)) = ∅))
7067, 693anbi23d 1442 . . . . . . . . . . . . 13 (𝑤 = (𝑣 × 𝑆) → ((𝑥 ∈ (𝑢 × 𝑆) ∧ 𝑦𝑤 ∧ ((𝑢 × 𝑆) ∩ 𝑤) = ∅) ↔ (𝑥 ∈ (𝑢 × 𝑆) ∧ 𝑦 ∈ (𝑣 × 𝑆) ∧ ((𝑢 × 𝑆) ∩ (𝑣 × 𝑆)) = ∅)))
7166, 70rspc2ev 3355 . . . . . . . . . . . 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)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
7234, 37, 47, 57, 62, 71syl113anc 1378 . . . . . . . . . . 11 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ ((𝑢𝑅𝑣𝑅) ∧ ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
7372expr 642 . . . . . . . . . 10 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) ∧ (𝑢𝑅𝑣𝑅)) → (((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅)))
7473rexlimdvva 3067 . . . . . . . . 9 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) → (∃𝑢𝑅𝑣𝑅 ((1st𝑥) ∈ 𝑢 ∧ (1st𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅)))
7526, 74mpd 15 . . . . . . . 8 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (1st𝑥) ≠ (1st𝑦)) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
76 simpllr 815 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) → 𝑆 ∈ Haus)
7743adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) → (2nd𝑥) ∈ 𝑆)
7853adantr 480 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) → (2nd𝑦) ∈ 𝑆)
79 simpr 476 . . . . . . . . . 10 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) → (2nd𝑥) ≠ (2nd𝑦))
806hausnei 21180 . . . . . . . . . 10 ((𝑆 ∈ Haus ∧ ((2nd𝑥) ∈ 𝑆 ∧ (2nd𝑦) ∈ 𝑆 ∧ (2nd𝑥) ≠ (2nd𝑦))) → ∃𝑢𝑆𝑣𝑆 ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))
8176, 77, 78, 79, 80syl13anc 1368 . . . . . . . . 9 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) → ∃𝑢𝑆𝑣𝑆 ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))
8227ad2antrr 762 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑅 ∈ Top)
832ad4antlr 771 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑆 ∈ Top)
845topopn 20759 . . . . . . . . . . . . . 14 (𝑅 ∈ Top → 𝑅𝑅)
8582, 84syl 17 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑅𝑅)
86 simprll 819 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑢𝑆)
87 txopn 21453 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ( 𝑅𝑅𝑢𝑆)) → ( 𝑅 × 𝑢) ∈ (𝑅 ×t 𝑆))
8882, 83, 85, 86, 87syl22anc 1367 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ( 𝑅 × 𝑢) ∈ (𝑅 ×t 𝑆))
89 simprlr 820 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑣𝑆)
90 txopn 21453 . . . . . . . . . . . . 13 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ( 𝑅𝑅𝑣𝑆)) → ( 𝑅 × 𝑣) ∈ (𝑅 ×t 𝑆))
9182, 83, 85, 89, 90syl22anc 1367 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ( 𝑅 × 𝑣) ∈ (𝑅 ×t 𝑆))
9239ad2antrr 762 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
9319ad2antrr 762 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (1st𝑥) ∈ 𝑅)
94 simprr1 1129 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (2nd𝑥) ∈ 𝑢)
9593, 94jca 553 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ((1st𝑥) ∈ 𝑅 ∧ (2nd𝑥) ∈ 𝑢))
96 elxp6 7244 . . . . . . . . . . . . 13 (𝑥 ∈ ( 𝑅 × 𝑢) ↔ (𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩ ∧ ((1st𝑥) ∈ 𝑅 ∧ (2nd𝑥) ∈ 𝑢)))
9792, 95, 96sylanbrc 699 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑥 ∈ ( 𝑅 × 𝑢))
9849ad2antrr 762 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
9922ad2antrr 762 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (1st𝑦) ∈ 𝑅)
100 simprr2 1130 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (2nd𝑦) ∈ 𝑣)
10199, 100jca 553 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ((1st𝑦) ∈ 𝑅 ∧ (2nd𝑦) ∈ 𝑣))
102 elxp6 7244 . . . . . . . . . . . . 13 (𝑦 ∈ ( 𝑅 × 𝑣) ↔ (𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩ ∧ ((1st𝑦) ∈ 𝑅 ∧ (2nd𝑦) ∈ 𝑣)))
10398, 101, 102sylanbrc 699 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → 𝑦 ∈ ( 𝑅 × 𝑣))
104 simprr3 1131 . . . . . . . . . . . . . 14 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (𝑢𝑣) = ∅)
105104xpeq2d 5173 . . . . . . . . . . . . 13 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ( 𝑅 × (𝑢𝑣)) = ( 𝑅 × ∅))
106 xpindi 5288 . . . . . . . . . . . . 13 ( 𝑅 × (𝑢𝑣)) = (( 𝑅 × 𝑢) ∩ ( 𝑅 × 𝑣))
107 xp0 5587 . . . . . . . . . . . . 13 ( 𝑅 × ∅) = ∅
108105, 106, 1073eqtr3g 2708 . . . . . . . . . . . 12 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → (( 𝑅 × 𝑢) ∩ ( 𝑅 × 𝑣)) = ∅)
109 eleq2 2719 . . . . . . . . . . . . . 14 (𝑧 = ( 𝑅 × 𝑢) → (𝑥𝑧𝑥 ∈ ( 𝑅 × 𝑢)))
110 ineq1 3840 . . . . . . . . . . . . . . 15 (𝑧 = ( 𝑅 × 𝑢) → (𝑧𝑤) = (( 𝑅 × 𝑢) ∩ 𝑤))
111110eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑧 = ( 𝑅 × 𝑢) → ((𝑧𝑤) = ∅ ↔ (( 𝑅 × 𝑢) ∩ 𝑤) = ∅))
112109, 1113anbi13d 1441 . . . . . . . . . . . . 13 (𝑧 = ( 𝑅 × 𝑢) → ((𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅) ↔ (𝑥 ∈ ( 𝑅 × 𝑢) ∧ 𝑦𝑤 ∧ (( 𝑅 × 𝑢) ∩ 𝑤) = ∅)))
113 eleq2 2719 . . . . . . . . . . . . . 14 (𝑤 = ( 𝑅 × 𝑣) → (𝑦𝑤𝑦 ∈ ( 𝑅 × 𝑣)))
114 ineq2 3841 . . . . . . . . . . . . . . 15 (𝑤 = ( 𝑅 × 𝑣) → (( 𝑅 × 𝑢) ∩ 𝑤) = (( 𝑅 × 𝑢) ∩ ( 𝑅 × 𝑣)))
115114eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑤 = ( 𝑅 × 𝑣) → ((( 𝑅 × 𝑢) ∩ 𝑤) = ∅ ↔ (( 𝑅 × 𝑢) ∩ ( 𝑅 × 𝑣)) = ∅))
116113, 1153anbi23d 1442 . . . . . . . . . . . . 13 (𝑤 = ( 𝑅 × 𝑣) → ((𝑥 ∈ ( 𝑅 × 𝑢) ∧ 𝑦𝑤 ∧ (( 𝑅 × 𝑢) ∩ 𝑤) = ∅) ↔ (𝑥 ∈ ( 𝑅 × 𝑢) ∧ 𝑦 ∈ ( 𝑅 × 𝑣) ∧ (( 𝑅 × 𝑢) ∩ ( 𝑅 × 𝑣)) = ∅)))
117112, 116rspc2ev 3355 . . . . . . . . . . . 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)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
11888, 91, 97, 103, 108, 117syl113anc 1378 . . . . . . . . . . 11 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ ((𝑢𝑆𝑣𝑆) ∧ ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅))) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
119118expr 642 . . . . . . . . . 10 (((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) ∧ (𝑢𝑆𝑣𝑆)) → (((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅)))
120119rexlimdvva 3067 . . . . . . . . 9 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) → (∃𝑢𝑆𝑣𝑆 ((2nd𝑥) ∈ 𝑢 ∧ (2nd𝑦) ∈ 𝑣 ∧ (𝑢𝑣) = ∅) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅)))
12181, 120mpd 15 . . . . . . . 8 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ (2nd𝑥) ≠ (2nd𝑦)) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
12275, 121jaodan 843 . . . . . . 7 ((((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) ∧ ((1st𝑥) ≠ (1st𝑦) ∨ (2nd𝑥) ≠ (2nd𝑦))) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))
123122ex 449 . . . . . 6 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (((1st𝑥) ≠ (1st𝑦) ∨ (2nd𝑥) ≠ (2nd𝑦)) → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅)))
12416, 123sylbird 250 . . . . 5 (((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) ∧ (𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆))) → (𝑥𝑦 → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅)))
125124ex 449 . . . 4 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ((𝑥 ∈ ( 𝑅 × 𝑆) ∧ 𝑦 ∈ ( 𝑅 × 𝑆)) → (𝑥𝑦 → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))))
12611, 125sylbird 250 . . 3 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ((𝑥 (𝑅 ×t 𝑆) ∧ 𝑦 (𝑅 ×t 𝑆)) → (𝑥𝑦 → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))))
127126ralrimivv 2999 . 2 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → ∀𝑥 (𝑅 ×t 𝑆)∀𝑦 (𝑅 ×t 𝑆)(𝑥𝑦 → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅)))
128 eqid 2651 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
129128ishaus 21174 . 2 ((𝑅 ×t 𝑆) ∈ Haus ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑥 (𝑅 ×t 𝑆)∀𝑦 (𝑅 ×t 𝑆)(𝑥𝑦 → ∃𝑧 ∈ (𝑅 ×t 𝑆)∃𝑤 ∈ (𝑅 ×t 𝑆)(𝑥𝑧𝑦𝑤 ∧ (𝑧𝑤) = ∅))))
1304, 127, 129sylanbrc 699 1 ((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×t 𝑆) ∈ Haus)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cin 3606  c0 3948  cop 4216   cuni 4468   × cxp 5141  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Topctop 20746  Hauscha 21160   ×t ctx 21411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-haus 21167  df-tx 21413
This theorem is referenced by: (None)
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