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Theorem wunfi 10128
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 3976 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ⊆ 𝑈))
4 eleq1 2903 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ∈ 𝑈))
53, 4imbi12d 348 . . . . 5 (𝑥 = ∅ → ((𝑥𝑈𝑥𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
65imbi2d 344 . . . 4 (𝑥 = ∅ → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7 sseq1 3976 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
8 eleq1 2903 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
97, 8imbi12d 348 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑈𝑥𝑈) ↔ (𝑦𝑈𝑦𝑈)))
109imbi2d 344 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝑦𝑈𝑦𝑈))))
11 sseq1 3976 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12 eleq1 2903 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ∈ 𝑈))
1311, 12imbi12d 348 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥𝑈𝑥𝑈) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
1413imbi2d 344 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
15 sseq1 3976 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
16 eleq1 2903 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
1715, 16imbi12d 348 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑈𝑥𝑈) ↔ (𝐴𝑈𝐴𝑈)))
1817imbi2d 344 . . . 4 (𝑥 = 𝐴 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝐴𝑈𝐴𝑈))))
19 wun0.1 . . . . . 6 (𝜑𝑈 ∈ WUni)
2019wun0 10125 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
2120a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22 ssun1 4132 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
23 sstr 3959 . . . . . . . . 9 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦𝑈)
2422, 23mpan 689 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)
2524imim1i 63 . . . . . . 7 ((𝑦𝑈𝑦𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈))
2619adantr 484 . . . . . . . . . 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 4148 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ⊆ 𝑈)
30 vex 3482 . . . . . . . . . . . . 13 𝑧 ∈ V
3130snss 4699 . . . . . . . . . . . 12 (𝑧𝑈 ↔ {𝑧} ⊆ 𝑈)
3229, 31sylibr 237 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑧𝑈)
3326, 32wunsn 10123 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ∈ 𝑈)
3426, 27, 33wunun 10117 . . . . . . . . 9 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
3534exp32 424 . . . . . . . 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 8742 . . 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 399   = wceq 1538  wcel 2115  cun 3916  wss 3918  c0 4274  {csn 4548  Fincfn 8492  WUnicwun 10107
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-1o 8085  df-er 8272  df-en 8493  df-fin 8496  df-wun 10109
This theorem is referenced by: (None)
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