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Theorem wunr1om 10750
Description: A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypothesis
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
Assertion
Ref Expression
wunr1om (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)

Proof of Theorem wunr1om
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2814 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
3 fveq2 6902 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2814 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1𝑦) ∈ 𝑈))
5 fveq2 6902 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2814 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
7 r10 9799 . . . . . . 7 (𝑅1‘∅) = ∅
8 wun0.1 . . . . . . . 8 (𝜑𝑈 ∈ WUni)
98wun0 10749 . . . . . . 7 (𝜑 → ∅ ∈ 𝑈)
107, 9eqeltrid 2833 . . . . . 6 (𝜑 → (𝑅1‘∅) ∈ 𝑈)
118adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝑈 ∈ WUni)
12 simpr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1𝑦) ∈ 𝑈)
1311, 12wunpw 10738 . . . . . . . 8 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝒫 (𝑅1𝑦) ∈ 𝑈)
14 nnon 7882 . . . . . . . . . 10 (𝑦 ∈ ω → 𝑦 ∈ On)
15 r1suc 9801 . . . . . . . . . 10 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1614, 15syl 17 . . . . . . . . 9 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1716eleq1d 2814 . . . . . . . 8 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑈))
1813, 17imbitrrid 245 . . . . . . 7 (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈))
1918expd 414 . . . . . 6 (𝑦 ∈ ω → (𝜑 → ((𝑅1𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈)))
202, 4, 6, 10, 19finds2 7912 . . . . 5 (𝑥 ∈ ω → (𝜑 → (𝑅1𝑥) ∈ 𝑈))
21 eleq1 2817 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑈𝑦𝑈))
2221imbi2d 339 . . . . 5 ((𝑅1𝑥) = 𝑦 → ((𝜑 → (𝑅1𝑥) ∈ 𝑈) ↔ (𝜑𝑦𝑈)))
2320, 22syl5ibcom 244 . . . 4 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈)))
2423rexlimiv 3145 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈))
25 r1fnon 9798 . . . . 5 𝑅1 Fn On
26 fnfun 6659 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2725, 26ax-mp 5 . . . 4 Fun 𝑅1
28 fvelima 6969 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2927, 28mpan 688 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
3024, 29syl11 33 . 2 (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑈))
3130ssrdv 3988 1 (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wrex 3067  wss 3949  c0 4326  𝒫 cpw 4606  cima 5685  Oncon0 6374  suc csuc 6376  Fun wfun 6547   Fn wfn 6548  cfv 6553  ωcom 7876  𝑅1cr1 9793  WUnicwun 10731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-r1 9795  df-wun 10733
This theorem is referenced by:  wunom  10751
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