Theorem wunfi 10477

Description: A weak universe contains all finite sets with elements drawn from the universe. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunfi.2	⊢ (𝜑 → 𝐴 ⊆ 𝑈)
wunfi.3	⊢ (𝜑 → 𝐴 ∈ Fin)

Assertion

Ref	Expression
wunfi	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Proof of Theorem wunfi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wunfi.2	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
2		wunfi.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
3		sseq1 3946	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑈 ↔ ∅ ⊆ 𝑈))
4		eleq1 2826	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑈 ↔ ∅ ∈ 𝑈))
5	3, 4	imbi12d 345	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
6	5	imbi2d 341	. . . 4 ⊢ (𝑥 = ∅ → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7		sseq1 3946	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑈 ↔ 𝑦 ⊆ 𝑈))
8		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑈 ↔ 𝑦 ∈ 𝑈))
9	7, 8	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈)))
10	9	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈))))
11		sseq1 3946	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ 𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12		eleq1 2826	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑈 ↔ (𝑦 ∪ {𝑧}) ∈ 𝑈))
13	11, 12	imbi12d 345	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
14	13	imbi2d 341	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
15		sseq1 3946	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑈 ↔ 𝐴 ⊆ 𝑈))
16		eleq1 2826	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
17	15, 16	imbi12d 345	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈)))
18	17	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))))
19		wun0.1	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
20	19	wun0 10474	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈)
21	20	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22		ssun1 4106	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
23		sstr 3929	. . . . . . . . 9 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦 ⊆ 𝑈)
24	22, 23	mpan 687	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ⊆ 𝑈)
25	24	imim1i 63	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ∈ 𝑈))
26	19	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ WUni)
27		simprr 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
28		simprl 768	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
29	28	unssbd 4122	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ⊆ 𝑈)
30		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
31	30	snss 4719	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑈 ↔ {𝑧} ⊆ 𝑈)
32	29, 31	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
33	26, 32	wunsn 10472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ∈ 𝑈)
34	26, 27, 33	wunun 10466	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
35	34	exp32 421	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∈ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
36	35	a2d 29	. . . . . . 7 ⊢ (𝜑 → (((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
37	25, 36	syl5 34	. . . . . 6 ⊢ (𝜑 → ((𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
38	37	a2i 14	. . . . 5 ⊢ ((𝜑 → (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
39	38	a1i 11	. . . 4 ⊢ (𝑦 ∈ Fin → ((𝜑 → (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
40	6, 10, 14, 18, 21, 39	findcard2 8947	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈)))
41	2, 40	mpcom 38	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))
42	1, 41	mpd 15	1 ⊢ (𝜑 → 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  Fincfn 8733  WUnicwun 10456

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-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

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-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  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-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  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-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-om 7713  df-en 8734  df-fin 8737  df-wun 10458

This theorem is referenced by: (None)