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Theorem wunfi 9727
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 3759 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ⊆ 𝑈))
4 eleq1 2819 . . . . . 6 (𝑥 = ∅ → (𝑥𝑈 ↔ ∅ ∈ 𝑈))
53, 4imbi12d 333 . . . . 5 (𝑥 = ∅ → ((𝑥𝑈𝑥𝑈) ↔ (∅ ⊆ 𝑈 → ∅ ∈ 𝑈)))
65imbi2d 329 . . . 4 (𝑥 = ∅ → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))))
7 sseq1 3759 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
8 eleq1 2819 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑈𝑦𝑈))
97, 8imbi12d 333 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑈𝑥𝑈) ↔ (𝑦𝑈𝑦𝑈)))
109imbi2d 329 . . . 4 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝑦𝑈𝑦𝑈))))
11 sseq1 3759 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝑈))
12 eleq1 2819 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝑈 ↔ (𝑦 ∪ {𝑧}) ∈ 𝑈))
1311, 12imbi12d 333 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥𝑈𝑥𝑈) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈)))
1413imbi2d 329 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → ((𝑦 ∪ {𝑧}) ⊆ 𝑈 → (𝑦 ∪ {𝑧}) ∈ 𝑈))))
15 sseq1 3759 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
16 eleq1 2819 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑈𝐴𝑈))
1715, 16imbi12d 333 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑈𝑥𝑈) ↔ (𝐴𝑈𝐴𝑈)))
1817imbi2d 329 . . . 4 (𝑥 = 𝐴 → ((𝜑 → (𝑥𝑈𝑥𝑈)) ↔ (𝜑 → (𝐴𝑈𝐴𝑈))))
19 wun0.1 . . . . . 6 (𝜑𝑈 ∈ WUni)
2019wun0 9724 . . . . 5 (𝜑 → ∅ ∈ 𝑈)
2120a1d 25 . . . 4 (𝜑 → (∅ ⊆ 𝑈 → ∅ ∈ 𝑈))
22 ssun1 3911 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
23 sstr 3744 . . . . . . . . 9 ((𝑦 ⊆ (𝑦 ∪ {𝑧}) ∧ (𝑦 ∪ {𝑧}) ⊆ 𝑈) → 𝑦𝑈)
2422, 23mpan 708 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)
2524imim1i 63 . . . . . . 7 ((𝑦𝑈𝑦𝑈) → ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈))
2619adantr 472 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑈 ∈ WUni)
27 simprr 813 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑦𝑈)
28 simprl 811 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ⊆ 𝑈)
2928unssbd 3926 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ⊆ 𝑈)
30 vex 3335 . . . . . . . . . . . . 13 𝑧 ∈ V
3130snss 4452 . . . . . . . . . . . 12 (𝑧𝑈 ↔ {𝑧} ⊆ 𝑈)
3229, 31sylibr 224 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → 𝑧𝑈)
3326, 32wunsn 9722 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → {𝑧} ∈ 𝑈)
3426, 27, 33wunun 9716 . . . . . . . . 9 ((𝜑 ∧ ((𝑦 ∪ {𝑧}) ⊆ 𝑈𝑦𝑈)) → (𝑦 ∪ {𝑧}) ∈ 𝑈)
3534exp32 632 . . . . . . . 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 8357 . . 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 383   = wceq 1624  wcel 2131  cun 3705  wss 3707  c0 4050  {csn 4313  Fincfn 8113  WUnicwun 9706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-1o 7721  df-er 7903  df-en 8114  df-fin 8117  df-wun 9708
This theorem is referenced by: (None)
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