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Theorem tskuni 10539

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 10512	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ 𝑇)
2		cardidg 10304	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ Tarski → (card‘𝑇) ≈ 𝑇)
3	2	ensymd 8791	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ Tarski → 𝑇 ≈ (card‘𝑇))
4	3	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝑇 ≈ (card‘𝑇))
5		sdomentr 8898	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝐴 ≺ (card‘𝑇))
6	1, 4, 5	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ (card‘𝑇))
7		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
8	7	rnmpt 5864	. . . . . . . . . . . . . 14 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
9		cardon 9702	. . . . . . . . . . . . . . . . 17 ⊢ (card‘𝑇) ∈ On
10		sdomdom 8768	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ≼ (card‘𝑇))
11		ondomen 9793	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑇) ∈ On ∧ 𝐴 ≼ (card‘𝑇)) → 𝐴 ∈ dom card)
12	9, 10, 11	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ∈ dom card)
13	12	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → 𝐴 ∈ dom card)
14		vex 3436	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
15	14	imaex 7763	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑥) ∈ V
16	15, 7	fnmpti 6576	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴
17		dffn4 6694	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)))
18	16, 17	mpbi 229	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
19		fodomnum 9813	. . . . . . . . . . . . . . 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	eqbrtrrid 5110	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴)
22		domsdomtr 8899	. . . . . . . . . . . . 13 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
23	21, 22	sylancom 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
24	23	adantll 711	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
25	6, 24	mpdan 684	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
26		ne0i 4268	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑇 → 𝑇 ≠ ∅)
27		tskcard 10537	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (card‘𝑇) ∈ Inacc)
28	26, 27	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
29		elina 10443	. . . . . . . . . . . 12 ⊢ ((card‘𝑇) ∈ Inacc ↔ ((card‘𝑇) ≠ ∅ ∧ (cf‘(card‘𝑇)) = (card‘𝑇) ∧ ∀𝑥 ∈ (card‘𝑇)𝒫 𝑥 ≺ (card‘𝑇)))
30	29	simp2bi 1145	. . . . . . . . . . 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 5102	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
33	32	3adant2 1130	. . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
34	33	adantr 481	. . . . . . 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 1130	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
36	35	adantr 481	. . . . . . . . 9 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (card‘𝑇) ∈ Inacc)
37		inawina 10446	. . . . . . . . 9 ⊢ ((card‘𝑇) ∈ Inacc → (card‘𝑇) ∈ Inacc_w)
38		winalim 10451	. . . . . . . . 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 3436	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
41		eqeq1 2742	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 = (𝑓 “ 𝑥) ↔ 𝑦 = (𝑓 “ 𝑥)))
42	41	rexbidv 3226	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥)))
43	40, 42	elab 3609	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥))
44		imassrn 5980	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
45		f1ofo 6723	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–onto→(card‘𝑇))
46		forn 6691	. . . . . . . . . . . . . . 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	sseqtrid 3973	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (𝑓 “ 𝑥) ⊆ (card‘𝑇))
49	48	ad2antlr 724	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ⊆ (card‘𝑇))
50		f1of1 6715	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–1-1→(card‘𝑇))
51		elssuni 4871	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
52		vex 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53	52	f1imaen 8802	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:∪ 𝐴–1-1→(card‘𝑇) ∧ 𝑥 ⊆ ∪ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
54	50, 51, 53	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
55	54	adantll 711	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
56		simpl1 1190	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑇 ∈ Tarski)
57		trss 5200	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → 𝐴 ⊆ 𝑇))
58	57	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
59	58	3adant1 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
60	59	sselda 3921	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑇)
61		tsksdom 10512	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → 𝑥 ≺ 𝑇)
62	56, 60, 61	syl2anc 584	. . . . . . . . . . . . . . . 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 8898	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝑥 ≺ (card‘𝑇))
65	62, 63, 64	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
66	65	adantlr 712	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
67		ensdomtr 8900	. . . . . . . . . . . . . 14 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≺ (card‘𝑇)) → (𝑓 “ 𝑥) ≺ (card‘𝑇))
68	55, 66, 67	syl2anc 584	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 5102	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))
72		sseq1 3946	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ↔ (𝑓 “ 𝑥) ⊆ (card‘𝑇)))
73		breq1 5077	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ≺ (cf‘(card‘𝑇)) ↔ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇))))
74	72, 73	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 “ 𝑥) → ((𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇))) ↔ ((𝑓 “ 𝑥) ⊆ (card‘𝑇) ∧ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))))
75	74	biimprcd 249	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 3213	. . . . . . . . . 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 241	. . . . . . . . 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 3102	. . . . . . . 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 6787	. . . . . . . . 9 ⊢ (card‘𝑇) ∈ V
81	80	cfslb2n 10024	. . . . . . . 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 584	. . . . . . 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 4963	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
85	48	ralrimivw 3104	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
86		iunss 4975	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇) ↔ ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
87	85, 86	sylibr 233	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
88		fof 6688	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → 𝑓:∪ 𝐴⟶(card‘𝑇))
89		foelrn 6982	. . . . . . . . . . . . 13 ⊢ ((𝑓:∪ 𝐴–onto→(card‘𝑇) ∧ 𝑦 ∈ (card‘𝑇)) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧))
90	89	ex 413	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧)))
91		eluni2 4843	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥)
92		nfv 1917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥 𝑓:∪ 𝐴⟶(card‘𝑇)
93		nfiu1 4958	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
94	93	nfel2 2925	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
95		ssiun2 4977	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
96	95	3ad2ant2 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
97		ffn 6600	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → 𝑓 Fn ∪ 𝐴)
98	97	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑓 Fn ∪ 𝐴)
99	51	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ∪ 𝐴)
100		simp3 1137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
101		fnfvima 7109	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn ∪ 𝐴 ∧ 𝑥 ⊆ ∪ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
102	98, 99, 100, 101	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
103	96, 102	sseldd 3922	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
104	103	3exp 1118	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))))
105	92, 94, 104	rexlimd 3250	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
106	91, 105	syl5bi 241	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑧 ∈ ∪ 𝐴 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
107		eleq1a 2834	. . . . . . . . . . . . . 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 3212	. . . . . . . . . . . 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 3927	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (card‘𝑇) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
113	87, 112	eqssd 3938	. . . . . . . 8 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = (card‘𝑇))
114	84, 113	eqtr3id 2792	. . . . . . 7 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} = (card‘𝑇))
115	114	necon3ai 2968	. . . . . 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 796	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
118	117	nexdv 1939	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
119		entr 8792	. . . . . . 7 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → ∪ 𝐴 ≈ (card‘𝑇))
120	3, 119	sylan2 593	. . . . . 6 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∪ 𝐴 ≈ (card‘𝑇))
121		bren 8743	. . . . . 6 ⊢ (∪ 𝐴 ≈ (card‘𝑇) ↔ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
122	120, 121	sylib 217	. . . . 5 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
123	122	expcom 414	. . . 4 ⊢ (𝑇 ∈ Tarski → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
124	123	3ad2ant1 1132	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
125	118, 124	mtod 197	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∪ 𝐴 ≈ 𝑇)
126		uniss 4847	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ ∪ 𝑇)
127		df-tr 5192	. . . . . . . . . 10 ⊢ (Tr 𝑇 ↔ ∪ 𝑇 ⊆ 𝑇)
128	127	biimpi 215	. . . . . . . . 9 ⊢ (Tr 𝑇 → ∪ 𝑇 ⊆ 𝑇)
129	126, 128	sylan9ss 3934	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑇 ∧ Tr 𝑇) → ∪ 𝐴 ⊆ 𝑇)
130	129	expcom 414	. . . . . . 7 ⊢ (Tr 𝑇 → (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
131	57, 130	syld 47	. . . . . 6 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
132	131	imp 407	. . . . 5 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ⊆ 𝑇)
133		tsken 10510	. . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ ∪ 𝐴 ⊆ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
134	132, 133	sylan2 593	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ (Tr 𝑇 ∧ 𝐴 ∈ 𝑇)) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
135	134	3impb 1114	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
136	135	ord 861	. 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 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 ∪ cuni 4839 ∪ ciun 4924 class class class wbr 5074 ↦ cmpt 5157 Tr wtr 5191 dom cdm 5589 ran crn 5590 “ cima 5592 Oncon0 6266 Lim wlim 6267 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 ≈ cen 8730 ≼ cdom 8731 ≺ csdm 8732 cardccrd 9693 cfccf 9695 Inacc_wcwina 10438 Inacccina 10439 Tarskictsk 10504

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-ac2 10219

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-smo 8177 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-oi 9269 df-har 9316 df-r1 9522 df-card 9697 df-aleph 9698 df-cf 9699 df-acn 9700 df-ac 9872 df-wina 10440 df-ina 10441 df-tsk 10505

This theorem is referenced by: tskwun 10540 tskint 10541 tskun 10542 tskurn 10545 pwinfi3 41170

W3C validator