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Theorem tskuni 10003

Description: The union of an element of a transitive Tarski class is in the set. (Contributed by Mario Carneiro, 22-Jun-2013.)

**Assertion**
Ref	Expression
tskuni	⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇)

Proof of Theorem tskuni Dummy variables 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tsksdom 9976	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ 𝑇)
2		cardidg 9768	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ Tarski → (card‘𝑇) ≈ 𝑇)
3	2	ensymd 8357	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ Tarski → 𝑇 ≈ (card‘𝑇))
4	3	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝑇 ≈ (card‘𝑇))
5		sdomentr 8447	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝐴 ≺ (card‘𝑇))
6	1, 4, 5	syl2anc 576	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ (card‘𝑇))
7		eqid 2778	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
8	7	rnmpt 5670	. . . . . . . . . . . . . 14 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
9		cardon 9167	. . . . . . . . . . . . . . . . 17 ⊢ (card‘𝑇) ∈ On
10		sdomdom 8334	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ≼ (card‘𝑇))
11		ondomen 9257	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑇) ∈ On ∧ 𝐴 ≼ (card‘𝑇)) → 𝐴 ∈ dom card)
12	9, 10, 11	sylancr 578	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ∈ dom card)
13	12	adantl 474	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → 𝐴 ∈ dom card)
14		vex 3418	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
15	14	imaex 7436	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑥) ∈ V
16	15, 7	fnmpti 6321	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴
17		dffn4 6425	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)))
18	16, 17	mpbi 222	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
19		fodomnum 9277	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom card → ((𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) → ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) ≼ 𝐴))
20	13, 18, 19	mpisyl 21	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) ≼ 𝐴)
21	8, 20	syl5eqbrr 4965	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴)
22		domsdomtr 8448	. . . . . . . . . . . . 13 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
23	21, 22	sylancom 579	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
24	23	adantll 701	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
25	6, 24	mpdan 674	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
26		ne0i 4186	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑇 → 𝑇 ≠ ∅)
27		tskcard 10001	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (card‘𝑇) ∈ Inacc)
28	26, 27	sylan2 583	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
29		elina 9907	. . . . . . . . . . . 12 ⊢ ((card‘𝑇) ∈ Inacc ↔ ((card‘𝑇) ≠ ∅ ∧ (cf‘(card‘𝑇)) = (card‘𝑇) ∧ ∀𝑥 ∈ (card‘𝑇)𝒫 𝑥 ≺ (card‘𝑇)))
30	29	simp2bi 1126	. . . . . . . . . . 11 ⊢ ((card‘𝑇) ∈ Inacc → (cf‘(card‘𝑇)) = (card‘𝑇))
31	28, 30	syl 17	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (cf‘(card‘𝑇)) = (card‘𝑇))
32	25, 31	breqtrrd 4957	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
33	32	3adant2 1111	. . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
34	33	adantr 473	. . . . . . 7 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
35	28	3adant2 1111	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
36	35	adantr 473	. . . . . . . . 9 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (card‘𝑇) ∈ Inacc)
37		inawina 9910	. . . . . . . . 9 ⊢ ((card‘𝑇) ∈ Inacc → (card‘𝑇) ∈ Inacc_w)
38		winalim 9915	. . . . . . . . 9 ⊢ ((card‘𝑇) ∈ Inacc_w → Lim (card‘𝑇))
39	36, 37, 38	3syl 18	. . . . . . . 8 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → Lim (card‘𝑇))
40		vex 3418	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
41		eqeq1 2782	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 = (𝑓 “ 𝑥) ↔ 𝑦 = (𝑓 “ 𝑥)))
42	41	rexbidv 3242	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥)))
43	40, 42	elab 3582	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥))
44		imassrn 5781	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
45		f1ofo 6451	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–onto→(card‘𝑇))
46		forn 6422	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → ran 𝑓 = (card‘𝑇))
47	45, 46	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ran 𝑓 = (card‘𝑇))
48	44, 47	syl5sseq 3909	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (𝑓 “ 𝑥) ⊆ (card‘𝑇))
49	48	ad2antlr 714	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ⊆ (card‘𝑇))
50		f1of1 6443	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–1-1→(card‘𝑇))
51		elssuni 4741	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
52		vex 3418	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53	52	f1imaen 8368	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:∪ 𝐴–1-1→(card‘𝑇) ∧ 𝑥 ⊆ ∪ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
54	50, 51, 53	syl2an 586	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
55	54	adantll 701	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
56		simpl1 1171	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑇 ∈ Tarski)
57		trss 5039	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → 𝐴 ⊆ 𝑇))
58	57	imp 398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
59	58	3adant1 1110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
60	59	sselda 3858	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑇)
61		tsksdom 9976	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → 𝑥 ≺ 𝑇)
62	56, 60, 61	syl2anc 576	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ 𝑇)
63	56, 3	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑇 ≈ (card‘𝑇))
64		sdomentr 8447	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝑥 ≺ (card‘𝑇))
65	62, 63, 64	syl2anc 576	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
66	65	adantlr 702	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
67		ensdomtr 8449	. . . . . . . . . . . . . 14 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≺ (card‘𝑇)) → (𝑓 “ 𝑥) ≺ (card‘𝑇))
68	55, 66, 67	syl2anc 576	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≺ (card‘𝑇))
69	36, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (cf‘(card‘𝑇)) = (card‘𝑇))
70	69	adantr 473	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (cf‘(card‘𝑇)) = (card‘𝑇))
71	68, 70	breqtrrd 4957	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))
72		sseq1 3882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ↔ (𝑓 “ 𝑥) ⊆ (card‘𝑇)))
73		breq1 4932	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ≺ (cf‘(card‘𝑇)) ↔ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇))))
74	72, 73	anbi12d 621	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 “ 𝑥) → ((𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇))) ↔ ((𝑓 “ 𝑥) ⊆ (card‘𝑇) ∧ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))))
75	74	biimprcd 242	. . . . . . . . . . . 12 ⊢ (((𝑓 “ 𝑥) ⊆ (card‘𝑇) ∧ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇))) → (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
76	49, 71, 75	syl2anc 576	. . . . . . . . . . 11 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
77	76	rexlimdva 3229	. . . . . . . . . 10 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
78	43, 77	syl5bi 234	. . . . . . . . 9 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} → (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))))
79	78	ralrimiv 3131	. . . . . . . 8 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇))))
80		fvex 6512	. . . . . . . . 9 ⊢ (card‘𝑇) ∈ V
81	80	cfslb2n 9488	. . . . . . . 8 ⊢ ((Lim (card‘𝑇) ∧ ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} (𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇)))) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇)))
82	39, 79, 81	syl2anc 576	. . . . . . 7 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇)))
83	34, 82	mpd 15	. . . . . 6 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇))
84	15	dfiun2 4828	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
85	48	ralrimivw 3133	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
86		iunss 4835	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇) ↔ ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
87	85, 86	sylibr 226	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
88		fof 6419	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → 𝑓:∪ 𝐴⟶(card‘𝑇))
89		foelrn 6695	. . . . . . . . . . . . 13 ⊢ ((𝑓:∪ 𝐴–onto→(card‘𝑇) ∧ 𝑦 ∈ (card‘𝑇)) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧))
90	89	ex 405	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧)))
91		eluni2 4716	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥)
92		nfv 1873	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥 𝑓:∪ 𝐴⟶(card‘𝑇)
93		nfiu1 4823	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
94	93	nfel2 2948	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
95		ssiun2 4837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
96	95	3ad2ant2 1114	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
97		ffn 6344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → 𝑓 Fn ∪ 𝐴)
98	97	3ad2ant1 1113	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑓 Fn ∪ 𝐴)
99	51	3ad2ant2 1114	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ∪ 𝐴)
100		simp3 1118	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
101		fnfvima 6820	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn ∪ 𝐴 ∧ 𝑥 ⊆ ∪ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
102	98, 99, 100, 101	syl3anc 1351	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
103	96, 102	sseldd 3859	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
104	103	3exp 1099	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))))
105	92, 94, 104	rexlimd 3260	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
106	91, 105	syl5bi 234	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑧 ∈ ∪ 𝐴 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
107		eleq1a 2861	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) → (𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
108	106, 107	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑧 ∈ ∪ 𝐴 → (𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))))
109	108	rexlimdv 3228	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
110	88, 90, 109	sylsyld 61	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
111	45, 110	syl 17	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
112	111	ssrdv 3864	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (card‘𝑇) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
113	87, 112	eqssd 3875	. . . . . . . 8 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = (card‘𝑇))
114	84, 113	syl5eqr 2828	. . . . . . 7 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} = (card‘𝑇))
115	114	necon3ai 2992	. . . . . 6 ⊢ (∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
116	83, 115	syl 17	. . . . 5 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
117	116	pm2.01da 786	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
118	117	nexdv 1895	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
119		entr 8358	. . . . . . 7 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → ∪ 𝐴 ≈ (card‘𝑇))
120	3, 119	sylan2 583	. . . . . 6 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∪ 𝐴 ≈ (card‘𝑇))
121		bren 8315	. . . . . 6 ⊢ (∪ 𝐴 ≈ (card‘𝑇) ↔ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
122	120, 121	sylib 210	. . . . 5 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
123	122	expcom 406	. . . 4 ⊢ (𝑇 ∈ Tarski → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
124	123	3ad2ant1 1113	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
125	118, 124	mtod 190	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∪ 𝐴 ≈ 𝑇)
126		uniss 4733	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ ∪ 𝑇)
127		df-tr 5031	. . . . . . . . . 10 ⊢ (Tr 𝑇 ↔ ∪ 𝑇 ⊆ 𝑇)
128	127	biimpi 208	. . . . . . . . 9 ⊢ (Tr 𝑇 → ∪ 𝑇 ⊆ 𝑇)
129	126, 128	sylan9ss 3871	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑇 ∧ Tr 𝑇) → ∪ 𝐴 ⊆ 𝑇)
130	129	expcom 406	. . . . . . 7 ⊢ (Tr 𝑇 → (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
131	57, 130	syld 47	. . . . . 6 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
132	131	imp 398	. . . . 5 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ⊆ 𝑇)
133		tsken 9974	. . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ ∪ 𝐴 ⊆ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
134	132, 133	sylan2 583	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ (Tr 𝑇 ∧ 𝐴 ∈ 𝑇)) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
135	134	3impb 1095	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
136	135	ord 850	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (¬ ∪ 𝐴 ≈ 𝑇 → ∪ 𝐴 ∈ 𝑇))
137	125, 136	mpd 15	1 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∨ wo 833 ∧ w3a 1068 = wceq 1507 ∃wex 1742 ∈ wcel 2050 {cab 2758 ≠ wne 2967 ∀wral 3088 ∃wrex 3089 ⊆ wss 3829 ∅c0 4178 𝒫 cpw 4422 ∪ cuni 4712 ∪ ciun 4792 class class class wbr 4929 ↦ cmpt 5008 Tr wtr 5030 dom cdm 5407 ran crn 5408 “ cima 5410 Oncon0 6029 Lim wlim 6030 Fn wfn 6183 ⟶wf 6184 –1-1→wf1 6185 –onto→wfo 6186 –1-1-onto→wf1o 6187 ‘cfv 6188 ≈ cen 8303 ≼ cdom 8304 ≺ csdm 8305 cardccrd 9158 cfccf 9160 Inacc_wcwina 9902 Inacccina 9903 Tarskictsk 9968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-ac2 9683

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-smo 7787 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-oi 8769 df-har 8817 df-r1 8987 df-card 9162 df-aleph 9163 df-cf 9164 df-acn 9165 df-ac 9336 df-wina 9904 df-ina 9905 df-tsk 9969

This theorem is referenced by: tskwun 10004 tskint 10005 tskun 10006 tskurn 10009 pwinfi3 39290

W3C validator