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Theorem wunfi 10758
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 4020 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ⊆ 𝑈))
4 eleq1 2826 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ∈ 𝑈))
53, 4imbi12d 344 . . . . 5 (𝑥 = ∅ → ((𝑥𝑈𝑥𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
65imbi2d 340 . . . 4 (𝑥 = ∅ → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7 sseq1 4020 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
8 eleq1 2826 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
97, 8imbi12d 344 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑈𝑥𝑈) ↔ (𝑦𝑈𝑦𝑈)))
109imbi2d 340 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝑦𝑈𝑦𝑈))))
11 sseq1 4020 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12 eleq1 2826 . . . . . 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 4020 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
16 eleq1 2826 . . . . . 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 10755 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
2120a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22 ssun1 4187 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
23 sstr 4003 . . . . . . . . 9 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦𝑈)
2422, 23mpan 690 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)
2524imim1i 63 . . . . . . 7 ((𝑦𝑈𝑦𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈))
2619adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑈 ∈ WUni)
27 simprr 773 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑦𝑈)
28 simprl 771 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
2928unssbd 4203 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ⊆ 𝑈)
30 vex 3481 . . . . . . . . . . . . 13 𝑧 ∈ V
3130snss 4789 . . . . . . . . . . . 12 (𝑧𝑈 ↔ {𝑧} ⊆ 𝑈)
3229, 31sylibr 234 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑧𝑈)
3326, 32wunsn 10753 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ∈ 𝑈)
3426, 27, 33wunun 10747 . . . . . . . . 9 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
3534exp32 420 . . . . . . . 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 9202 . . 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 395   = wceq 1536  wcel 2105  cun 3960  wss 3962  c0 4338  {csn 4630  Fincfn 8983  WUnicwun 10737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-en 8984  df-fin 8987  df-wun 10739
This theorem is referenced by: (None)
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