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Theorem tskuni 10792

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 10765	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ 𝑇)
2		cardidg 10556	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ Tarski → (card‘𝑇) ≈ 𝑇)
3	2	ensymd 9011	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ Tarski → 𝑇 ≈ (card‘𝑇))
4	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝑇 ≈ (card‘𝑇))
5		sdomentr 9109	. . . . . . . . . . . 12 ⊢ ((𝐴 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝐴 ≺ (card‘𝑇))
6	1, 4, 5	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → 𝐴 ≺ (card‘𝑇))
7		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
8	7	rnmpt 5941	. . . . . . . . . . . . . 14 ⊢ ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
9		cardon 9949	. . . . . . . . . . . . . . . . 17 ⊢ (card‘𝑇) ∈ On
10		sdomdom 8986	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ≼ (card‘𝑇))
11		ondomen 10040	. . . . . . . . . . . . . . . . 17 ⊢ (((card‘𝑇) ∈ On ∧ 𝐴 ≼ (card‘𝑇)) → 𝐴 ∈ dom card)
12	9, 10, 11	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ≺ (card‘𝑇) → 𝐴 ∈ dom card)
13	12	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → 𝐴 ∈ dom card)
14		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
15	14	imaex 7911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑥) ∈ V
16	15, 7	fnmpti 6675	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴
17		dffn4 6795	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)))
18	16, 17	mpbi 233	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)):𝐴–onto→ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥))
19		fodomnum 10060	. . . . . . . . . . . . . . 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 22	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → ran (𝑥 ∈ 𝐴 ↦ (𝑓 “ 𝑥)) ≼ 𝐴)
21	8, 20	eqbrtrrid 5141	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴)
22		domsdomtr 9110	. . . . . . . . . . . . 13 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≼ 𝐴 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
23	21, 22	sylancom 600	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
24	23	adantll 727	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) ∧ 𝐴 ≺ (card‘𝑇)) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
25	6, 24	mpdan 700	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (card‘𝑇))
26		ne0i 4287	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑇 → 𝑇 ≠ ∅)
27		tskcard 10790	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → (card‘𝑇) ∈ Inacc)
28	26, 27	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
29		elina 10696	. . . . . . . . . . . 12 ⊢ ((card‘𝑇) ∈ Inacc ↔ ((card‘𝑇) ≠ ∅ ∧ (cf‘(card‘𝑇)) = (card‘𝑇) ∧ ∀𝑥 ∈ (card‘𝑇)𝒫 𝑥 ≺ (card‘𝑇)))
30	29	simp2bi 1164	. . . . . . . . . . 11 ⊢ ((card‘𝑇) ∈ Inacc → (cf‘(card‘𝑇)) = (card‘𝑇))
31	28, 30	syl 18	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → (cf‘(card‘𝑇)) = (card‘𝑇))
32	25, 31	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
33	32	3adant2 1149	. . . . . . . 8 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≺ (cf‘(card‘𝑇)))
34	33	adantr 486	. . . . . . 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 1149	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (card‘𝑇) ∈ Inacc)
36	35	adantr 486	. . . . . . . . 9 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (card‘𝑇) ∈ Inacc)
37		inawina 10699	. . . . . . . . 9 ⊢ ((card‘𝑇) ∈ Inacc → (card‘𝑇) ∈ Inacc_w)
38		winalim 10704	. . . . . . . . 9 ⊢ ((card‘𝑇) ∈ Inacc_w → Lim (card‘𝑇))
39	36, 37, 38	3syl 19	. . . . . . . 8 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → Lim (card‘𝑇))
40		vex 3454	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
41		eqeq1 2764	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → (𝑧 = (𝑓 “ 𝑥) ↔ 𝑦 = (𝑓 “ 𝑥)))
42	41	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥)))
43	40, 42	elab 3633	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝑓 “ 𝑥))
44		imassrn 6067	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
45		f1ofo 6825	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–onto→(card‘𝑇))
46		forn 6792	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → ran 𝑓 = (card‘𝑇))
47	45, 46	syl 18	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ⊆ (card‘𝑇))
50		f1of1 6816	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → 𝑓:∪ 𝐴–1-1→(card‘𝑇))
51		elssuni 4899	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
52		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
53	52	f1imaen 9023	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:∪ 𝐴–1-1→(card‘𝑇) ∧ 𝑥 ⊆ ∪ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
54	50, 51, 53	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
55	54	adantll 727	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝑥)
56		simpl1 1210	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑇 ∈ Tarski)
57		trss 5222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → 𝐴 ⊆ 𝑇))
58	57	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
59	58	3adant1 1148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → 𝐴 ⊆ 𝑇)
60	59	sselda 3931	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑇)
61		tsksdom 10765	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇 ∈ Tarski ∧ 𝑥 ∈ 𝑇) → 𝑥 ≺ 𝑇)
62	56, 60, 61	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ 𝑇)
63	56, 3	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑇 ≈ (card‘𝑇))
64		sdomentr 9109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ≺ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → 𝑥 ≺ (card‘𝑇))
65	62, 63, 64	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
66	65	adantlr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≺ (card‘𝑇))
67		ensdomtr 9111	. . . . . . . . . . . . . 14 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≺ (card‘𝑇)) → (𝑓 “ 𝑥) ≺ (card‘𝑇))
68	55, 66, 67	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≺ (card‘𝑇))
69	36, 30	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) → (cf‘(card‘𝑇)) = (card‘𝑇))
70	69	adantr 486	. . . . . . . . . . . . 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 5133	. . . . . . . . . . . 12 ⊢ ((((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) ∧ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)) ∧ 𝑥 ∈ 𝐴) → (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))
72		sseq1 3956	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ⊆ (card‘𝑇) ↔ (𝑓 “ 𝑥) ⊆ (card‘𝑇)))
73		breq1 5106	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑓 “ 𝑥) → (𝑦 ≺ (cf‘(card‘𝑇)) ↔ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇))))
74	72, 73	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑓 “ 𝑥) → ((𝑦 ⊆ (card‘𝑇) ∧ 𝑦 ≺ (cf‘(card‘𝑇))) ↔ ((𝑓 “ 𝑥) ⊆ (card‘𝑇) ∧ (𝑓 “ 𝑥) ≺ (cf‘(card‘𝑇)))))
75	74	biimprcd 253	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 3163	. . . . . . . . . 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	biimtrid 245	. . . . . . . . 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 3153	. . . . . . . 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 6891	. . . . . . . . 9 ⊢ (card‘𝑇) ∈ V
81	80	cfslb2n 10270	. . . . . . . 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 596	. . . . . . 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 16	. . . . . 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 4990	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)}
85	48	ralrimivw 3158	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
86		iunss 5003	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇) ↔ ∀𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
87	85, 86	sylibr 237	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) ⊆ (card‘𝑇))
88		fof 6789	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → 𝑓:∪ 𝐴⟶(card‘𝑇))
89		foelrn 7100	. . . . . . . . . . . . 13 ⊢ ((𝑓:∪ 𝐴–onto→(card‘𝑇) ∧ 𝑦 ∈ (card‘𝑇)) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧))
90	89	ex 418	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → ∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧)))
91		eluni2 4871	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥)
92		nfv 1947	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥 𝑓:∪ 𝐴⟶(card‘𝑇)
93		nfiu1 4986	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
94	93	nfel2 2940	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)
95		ssiun2 5006	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
96	95	3ad2ant2 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓 “ 𝑥) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
97		ffn 6702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → 𝑓 Fn ∪ 𝐴)
98	97	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑓 Fn ∪ 𝐴)
99	51	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑥 ⊆ ∪ 𝐴)
100		simp3 1156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ 𝑥)
101		fnfvima 7232	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn ∪ 𝐴 ∧ 𝑥 ⊆ ∪ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
102	98, 99, 100, 101	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ (𝑓 “ 𝑥))
103	96, 102	sseldd 3932	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:∪ 𝐴⟶(card‘𝑇) ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝑥) → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
104	103	3exp 1137	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))))
105	92, 94, 104	rexlimd 3269	. . . . . . . . . . . . . . 15 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝑥 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
106	91, 105	biimtrid 245	. . . . . . . . . . . . . 14 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑧 ∈ ∪ 𝐴 → (𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
107		eleq1a 2855	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑧) ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) → (𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
108	106, 107	syl6 36	. . . . . . . . . . . . 13 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (𝑧 ∈ ∪ 𝐴 → (𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))))
109	108	rexlimdv 3161	. . . . . . . . . . . 12 ⊢ (𝑓:∪ 𝐴⟶(card‘𝑇) → (∃𝑧 ∈ ∪ 𝐴𝑦 = (𝑓‘𝑧) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
110	88, 90, 109	sylsyld 62	. . . . . . . . . . 11 ⊢ (𝑓:∪ 𝐴–onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
111	45, 110	syl 18	. . . . . . . . . 10 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (𝑦 ∈ (card‘𝑇) → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥)))
112	111	ssrdv 3937	. . . . . . . . 9 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → (card‘𝑇) ⊆ ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥))
113	87, 112	eqssd 3948	. . . . . . . 8 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ 𝑥 ∈ 𝐴 (𝑓 “ 𝑥) = (card‘𝑇))
114	84, 113	eqtr3id 2809	. . . . . . 7 ⊢ (𝑓:∪ 𝐴–1-1-onto→(card‘𝑇) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} = (card‘𝑇))
115	114	necon3ai 2980	. . . . . 6 ⊢ (∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (𝑓 “ 𝑥)} ≠ (card‘𝑇) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
116	83, 115	syl 18	. . . . 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 811	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
118	117	nexdv 1969	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
119		entr 9012	. . . . . . 7 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ≈ (card‘𝑇)) → ∪ 𝐴 ≈ (card‘𝑇))
120	3, 119	sylan2 605	. . . . . 6 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∪ 𝐴 ≈ (card‘𝑇))
121		bren 8962	. . . . . 6 ⊢ (∪ 𝐴 ≈ (card‘𝑇) ↔ ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
122	120, 121	sylib 221	. . . . 5 ⊢ ((∪ 𝐴 ≈ 𝑇 ∧ 𝑇 ∈ Tarski) → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇))
123	122	expcom 419	. . . 4 ⊢ (𝑇 ∈ Tarski → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
124	123	3ad2ant1 1151	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 → ∃𝑓 𝑓:∪ 𝐴–1-1-onto→(card‘𝑇)))
125	118, 124	mtod 201	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ¬ ∪ 𝐴 ≈ 𝑇)
126		uniss 4875	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ ∪ 𝑇)
127		df-tr 5213	. . . . . . . . . 10 ⊢ (Tr 𝑇 ↔ ∪ 𝑇 ⊆ 𝑇)
128	127	biimpi 219	. . . . . . . . 9 ⊢ (Tr 𝑇 → ∪ 𝑇 ⊆ 𝑇)
129	126, 128	sylan9ss 3944	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑇 ∧ Tr 𝑇) → ∪ 𝐴 ⊆ 𝑇)
130	129	expcom 419	. . . . . . 7 ⊢ (Tr 𝑇 → (𝐴 ⊆ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
131	57, 130	syld 48	. . . . . 6 ⊢ (Tr 𝑇 → (𝐴 ∈ 𝑇 → ∪ 𝐴 ⊆ 𝑇))
132	131	imp 412	. . . . 5 ⊢ ((Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ⊆ 𝑇)
133		tsken 10763	. . . . 5 ⊢ ((𝑇 ∈ Tarski ∧ ∪ 𝐴 ⊆ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
134	132, 133	sylan2 605	. . . 4 ⊢ ((𝑇 ∈ Tarski ∧ (Tr 𝑇 ∧ 𝐴 ∈ 𝑇)) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
135	134	3impb 1132	. . 3 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (∪ 𝐴 ≈ 𝑇 ∨ ∪ 𝐴 ∈ 𝑇))
136	135	ord 878	. 2 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → (¬ ∪ 𝐴 ≈ 𝑇 → ∪ 𝐴 ∈ 𝑇))
137	125, 136	mpd 16	1 ⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ 𝐴 ∈ 𝑇) → ∪ 𝐴 ∈ 𝑇)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2145 {cab 2738 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3899 ∅c0 4279 𝒫 cpw 4557 ∪ cuni 4867 ∪ ciun 4951 class class class wbr 5103 ↦ cmpt 5186 Tr wtr 5212 dom cdm 5655 ran crn 5656 “ cima 5658 Oncon0 6357 Lim wlim 6358 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 ≈ cen 8949 ≼ cdom 8950 ≺ csdm 8951 cardccrd 9940 cfccf 9942 Inacc_wcwina 10691 Inacccina 10692 Tarskictsk 10757

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-ac2 10465

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-smo 8335 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-oi 9482 df-har 9529 df-r1 9746 df-card 9944 df-aleph 9945 df-cf 9946 df-acn 9947 df-ac 10119 df-wina 10693 df-ina 10694 df-tsk 10758

This theorem is used by: tskwun 10793 tskint 10794 tskun 10795 tskurn 10798 pwinfi3 44403

W3C validator