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Theorem wunr1om 10129
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 6663 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2894 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
3 fveq2 6663 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2894 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1𝑦) ∈ 𝑈))
5 fveq2 6663 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2894 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
7 r10 9185 . . . . . . 7 (𝑅1‘∅) = ∅
8 wun0.1 . . . . . . . 8 (𝜑𝑈 ∈ WUni)
98wun0 10128 . . . . . . 7 (𝜑 → ∅ ∈ 𝑈)
107, 9eqeltrid 2914 . . . . . 6 (𝜑 → (𝑅1‘∅) ∈ 𝑈)
118adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝑈 ∈ WUni)
12 simpr 485 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1𝑦) ∈ 𝑈)
1311, 12wunpw 10117 . . . . . . . 8 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝒫 (𝑅1𝑦) ∈ 𝑈)
14 nnon 7575 . . . . . . . . . 10 (𝑦 ∈ ω → 𝑦 ∈ On)
15 r1suc 9187 . . . . . . . . . 10 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1614, 15syl 17 . . . . . . . . 9 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1716eleq1d 2894 . . . . . . . 8 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑈))
1813, 17syl5ibr 247 . . . . . . 7 (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈))
1918expd 416 . . . . . 6 (𝑦 ∈ ω → (𝜑 → ((𝑅1𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈)))
202, 4, 6, 10, 19finds2 7599 . . . . 5 (𝑥 ∈ ω → (𝜑 → (𝑅1𝑥) ∈ 𝑈))
21 eleq1 2897 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑈𝑦𝑈))
2221imbi2d 342 . . . . 5 ((𝑅1𝑥) = 𝑦 → ((𝜑 → (𝑅1𝑥) ∈ 𝑈) ↔ (𝜑𝑦𝑈)))
2320, 22syl5ibcom 246 . . . 4 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈)))
2423rexlimiv 3277 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈))
25 r1fnon 9184 . . . . 5 𝑅1 Fn On
26 fnfun 6446 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2725, 26ax-mp 5 . . . 4 Fun 𝑅1
28 fvelima 6724 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2927, 28mpan 686 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
3024, 29syl11 33 . 2 (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑈))
3130ssrdv 3970 1 (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  wss 3933  c0 4288  𝒫 cpw 4535  cima 5551  Oncon0 6184  suc csuc 6186  Fun wfun 6342   Fn wfn 6343  cfv 6348  ωcom 7569  𝑅1cr1 9179  WUnicwun 10110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-r1 9181  df-wun 10112
This theorem is referenced by:  wunom  10130
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