Theorem wunfi 10142

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 3991	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑈 ↔ ∅ ⊆ 𝑈))
4		eleq1 2900	. . . . . 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 3991	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑈 ↔ 𝑦 ⊆ 𝑈))
8		eleq1 2900	. . . . . 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 3991	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ 𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12		eleq1 2900	. . . . . 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 3991	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑈 ↔ 𝐴 ⊆ 𝑈))
16		eleq1 2900	. . . . . 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 10139	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈)
21	20	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22		ssun1 4147	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
23		sstr 3974	. . . . . . . . 9 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦 ⊆ 𝑈)
24	22, 23	mpan 688	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ⊆ 𝑈)
25	24	imim1i 63	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ∈ 𝑈))
26	19	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ WUni)
27		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
28		simprl 769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
29	28	unssbd 4163	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ⊆ 𝑈)
30		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
31	30	snss 4717	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑈 ↔ {𝑧} ⊆ 𝑈)
32	29, 31	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
33	26, 32	wunsn 10137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ∈ 𝑈)
34	26, 27, 33	wunun 10131	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
35	34	exp32 423	. . . . . . . 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 8757	. . 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 398   = wceq 1533   ∈ wcel 2110   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  {csn 4566  Fincfn 8508  WUnicwun 10121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-om 7580  df-1o 8101  df-er 8288  df-en 8509  df-fin 8512  df-wun 10123

This theorem is referenced by: (None)