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Theorem wunfi 10715
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 (𝜑𝑈 ∈ WUni)
wunfi.2 (𝜑𝐴𝑈)
wunfi.3 (𝜑𝐴 ∈ Fin)
Assertion
Ref Expression
wunfi (𝜑𝐴𝑈)

Proof of Theorem wunfi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wunfi.2 . 2 (𝜑𝐴𝑈)
2 wunfi.3 . . 3 (𝜑𝐴 ∈ Fin)
3 sseq1 4007 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ⊆ 𝑈))
4 eleq1 2821 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ∈ 𝑈))
53, 4imbi12d 344 . . . . 5 (𝑥 = ∅ → ((𝑥𝑈𝑥𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
65imbi2d 340 . . . 4 (𝑥 = ∅ → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7 sseq1 4007 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
8 eleq1 2821 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
97, 8imbi12d 344 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑈𝑥𝑈) ↔ (𝑦𝑈𝑦𝑈)))
109imbi2d 340 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝑦𝑈𝑦𝑈))))
11 sseq1 4007 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12 eleq1 2821 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ∈ 𝑈))
1311, 12imbi12d 344 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥𝑈𝑥𝑈) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
1413imbi2d 340 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
15 sseq1 4007 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
16 eleq1 2821 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
1715, 16imbi12d 344 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑈𝑥𝑈) ↔ (𝐴𝑈𝐴𝑈)))
1817imbi2d 340 . . . 4 (𝑥 = 𝐴 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝐴𝑈𝐴𝑈))))
19 wun0.1 . . . . . 6 (𝜑𝑈 ∈ WUni)
2019wun0 10712 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
2120a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22 ssun1 4172 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
23 sstr 3990 . . . . . . . . 9 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦𝑈)
2422, 23mpan 688 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)
2524imim1i 63 . . . . . . 7 ((𝑦𝑈𝑦𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈))
2619adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑈 ∈ WUni)
27 simprr 771 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑦𝑈)
28 simprl 769 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
2928unssbd 4188 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ⊆ 𝑈)
30 vex 3478 . . . . . . . . . . . . 13 𝑧 ∈ V
3130snss 4789 . . . . . . . . . . . 12 (𝑧𝑈 ↔ {𝑧} ⊆ 𝑈)
3229, 31sylibr 233 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑧𝑈)
3326, 32wunsn 10710 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ∈ 𝑈)
3426, 27, 33wunun 10704 . . . . . . . . 9 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
3534exp32 421 . . . . . . . 8 (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
3635a2d 29 . . . . . . 7 (𝜑 → (((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
3725, 36syl5 34 . . . . . 6 (𝜑 → ((𝑦𝑈𝑦𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
3837a2i 14 . . . . 5 ((𝜑 → (𝑦𝑈𝑦𝑈)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
3938a1i 11 . . . 4 (𝑦 ∈ Fin → ((𝜑 → (𝑦𝑈𝑦𝑈)) → (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
406, 10, 14, 18, 21, 39findcard2 9163 . . 3 (𝐴 ∈ Fin → (𝜑 → (𝐴𝑈𝐴𝑈)))
412, 40mpcom 38 . 2 (𝜑 → (𝐴𝑈𝐴𝑈))
421, 41mpd 15 1 (𝜑𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cun 3946  wss 3948  c0 4322  {csn 4628  Fincfn 8938  WUnicwun 10694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7855  df-en 8939  df-fin 8942  df-wun 10696
This theorem is referenced by: (None)
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