Theorem wunfi 9859

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 3852	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝑈 ↔ ∅ ⊆ 𝑈))
4		eleq1 2895	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑈 ↔ ∅ ∈ 𝑈))
5	3, 4	imbi12d 336	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
6	5	imbi2d 332	. . . 4 ⊢ (𝑥 = ∅ → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7		sseq1 3852	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝑈 ↔ 𝑦 ⊆ 𝑈))
8		eleq1 2895	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑈 ↔ 𝑦 ∈ 𝑈))
9	7, 8	imbi12d 336	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈)))
10	9	imbi2d 332	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈))))
11		sseq1 3852	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ 𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12		eleq1 2895	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ 𝑈 ↔ (𝑦 ∪ {𝑧}) ∈ 𝑈))
13	11, 12	imbi12d 336	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
14	13	imbi2d 332	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
15		sseq1 3852	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑈 ↔ 𝐴 ⊆ 𝑈))
16		eleq1 2895	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
17	15, 16	imbi12d 336	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈) ↔ (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈)))
18	17	imbi2d 332	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 → (𝑥 ⊆ 𝑈 → 𝑥 ∈ 𝑈)) ↔ (𝜑 → (𝐴 ⊆ 𝑈 → 𝐴 ∈ 𝑈))))
19		wun0.1	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
20	19	wun0 9856	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝑈)
21	20	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22		ssun1 4004	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
23		sstr 3836	. . . . . . . . 9 ⊢ ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦 ⊆ 𝑈)
24	22, 23	mpan 683	. . . . . . . 8 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ⊆ 𝑈)
25	24	imim1i 63	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝑈 → 𝑦 ∈ 𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → 𝑦 ∈ 𝑈))
26	19	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑈 ∈ WUni)
27		simprr 791	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑦 ∈ 𝑈)
28		simprl 789	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
29	28	unssbd 4019	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ⊆ 𝑈)
30		vex 3418	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
31	30	snss 4536	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑈 ↔ {𝑧} ⊆ 𝑈)
32	29, 31	sylibr 226	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝑧 ∈ 𝑈)
33	26, 32	wunsn 9854	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → {𝑧} ∈ 𝑈)
34	26, 27, 33	wunun 9848	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 ∧ 𝑦 ∈ 𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
35	34	exp32 413	. . . . . . . 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 8470	. . 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 386   = wceq 1658   ∈ wcel 2166   ∪ cun 3797   ⊆ wss 3799  ∅c0 4145  {csn 4398  Fincfn 8223  WUnicwun 9838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-om 7328  df-1o 7827  df-er 8010  df-en 8224  df-fin 8227  df-wun 9840

This theorem is referenced by: (None)