Theorem wunfi 10807

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 3933	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑈 ↔ ∅ ⊆ 𝑈))
4		eleq1 2848	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑈 ↔ ∅ ∈ 𝑈))
5	3, 4	imbi12d 347	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
6	5	imbi2d 343	. . . 4 ⊢ (𝑥 = ∅ → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7		sseq1 3933	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑈 ↔ 𝑦 ⊆ 𝑈))
8		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑈 ↔ 𝑦 ∈ 𝑈))
9	7, 8	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈)))
10	9	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈))))
11		sseq1 3933	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ 𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12		eleq1 2848	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑈 ↔ (𝑦 ∪ {𝑧}) ∈ 𝑈))
13	11, 12	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
14	13	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
15		sseq1 3933	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑈 ↔ 𝐴 ⊆ 𝑈))
16		eleq1 2848	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
17	15, 16	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈)))
18	17	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))))
19		wun0.1	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
20	19	wun0 10804	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈)
21	20	a1d 26	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22		ssun1 4176	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
23		sstr 3916	. . . . . . . . 9 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦 ⊆ 𝑈)
24	22, 23	mpan 703	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ⊆ 𝑈)
25	24	imim1i 64	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ∈ 𝑈))
26	19	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ WUni)
27		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
28		simprl 783	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
29	28	unssbd 4192	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ⊆ 𝑈)
30		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
31	30	snss 4744	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑈 ↔ {𝑧} ⊆ 𝑈)
32	29, 31	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
33	26, 32	wunsn 10802	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ∈ 𝑈)
34	26, 27, 33	wunun 10796	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
35	34	exp32 426	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∈ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
36	35	a2d 30	. . . . . . 7 ⊢ (𝜑 → (((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
37	25, 36	syl5 35	. . . . . 6 ⊢ (𝜑 → ((𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
38	37	a2i 15	. . . . 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 9181	. . 3 ⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈)))
41	2, 40	mpcom 39	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))
42	1, 41	mpd 16	1 ⊢ (𝜑 → 𝐴 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583  Fincfn 8974  WUnicwun 10786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-en 8975  df-fin 8978  df-wun 10788

This theorem is used by: (None)