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Theorem wunfi 10716
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 4008 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ⊆ 𝑈))
4 eleq1 2822 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ∈ 𝑈))
53, 4imbi12d 345 . . . . 5 (𝑥 = ∅ → ((𝑥𝑈𝑥𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
65imbi2d 341 . . . 4 (𝑥 = ∅ → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7 sseq1 4008 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
8 eleq1 2822 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
97, 8imbi12d 345 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑈𝑥𝑈) ↔ (𝑦𝑈𝑦𝑈)))
109imbi2d 341 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝑦𝑈𝑦𝑈))))
11 sseq1 4008 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12 eleq1 2822 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ∈ 𝑈))
1311, 12imbi12d 345 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥𝑈𝑥𝑈) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
1413imbi2d 341 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
15 sseq1 4008 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
16 eleq1 2822 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
1715, 16imbi12d 345 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑈𝑥𝑈) ↔ (𝐴𝑈𝐴𝑈)))
1817imbi2d 341 . . . 4 (𝑥 = 𝐴 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝐴𝑈𝐴𝑈))))
19 wun0.1 . . . . . 6 (𝜑𝑈 ∈ WUni)
2019wun0 10713 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
2120a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22 ssun1 4173 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
23 sstr 3991 . . . . . . . . 9 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦𝑈)
2422, 23mpan 689 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)
2524imim1i 63 . . . . . . 7 ((𝑦𝑈𝑦𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈))
2619adantr 482 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑈 ∈ WUni)
27 simprr 772 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑦𝑈)
28 simprl 770 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
2928unssbd 4189 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ⊆ 𝑈)
30 vex 3479 . . . . . . . . . . . . 13 𝑧 ∈ V
3130snss 4790 . . . . . . . . . . . 12 (𝑧𝑈 ↔ {𝑧} ⊆ 𝑈)
3229, 31sylibr 233 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑧𝑈)
3326, 32wunsn 10711 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ∈ 𝑈)
3426, 27, 33wunun 10705 . . . . . . . . 9 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
3534exp32 422 . . . . . . . 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 9164 . . 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 397   = wceq 1542  wcel 2107  cun 3947  wss 3949  c0 4323  {csn 4629  Fincfn 8939  WUnicwun 10695
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-en 8940  df-fin 8943  df-wun 10697
This theorem is referenced by: (None)
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