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Theorem wunr1om 10656
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 6843 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2823 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
3 fveq2 6843 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2823 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1𝑦) ∈ 𝑈))
5 fveq2 6843 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2823 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
7 r10 9705 . . . . . . 7 (𝑅1‘∅) = ∅
8 wun0.1 . . . . . . . 8 (𝜑𝑈 ∈ WUni)
98wun0 10655 . . . . . . 7 (𝜑 → ∅ ∈ 𝑈)
107, 9eqeltrid 2842 . . . . . 6 (𝜑 → (𝑅1‘∅) ∈ 𝑈)
118adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝑈 ∈ WUni)
12 simpr 486 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1𝑦) ∈ 𝑈)
1311, 12wunpw 10644 . . . . . . . 8 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝒫 (𝑅1𝑦) ∈ 𝑈)
14 nnon 7809 . . . . . . . . . 10 (𝑦 ∈ ω → 𝑦 ∈ On)
15 r1suc 9707 . . . . . . . . . 10 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1614, 15syl 17 . . . . . . . . 9 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1716eleq1d 2823 . . . . . . . 8 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑈))
1813, 17syl5ibr 246 . . . . . . 7 (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈))
1918expd 417 . . . . . 6 (𝑦 ∈ ω → (𝜑 → ((𝑅1𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈)))
202, 4, 6, 10, 19finds2 7838 . . . . 5 (𝑥 ∈ ω → (𝜑 → (𝑅1𝑥) ∈ 𝑈))
21 eleq1 2826 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑈𝑦𝑈))
2221imbi2d 341 . . . . 5 ((𝑅1𝑥) = 𝑦 → ((𝜑 → (𝑅1𝑥) ∈ 𝑈) ↔ (𝜑𝑦𝑈)))
2320, 22syl5ibcom 244 . . . 4 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈)))
2423rexlimiv 3146 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈))
25 r1fnon 9704 . . . . 5 𝑅1 Fn On
26 fnfun 6603 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2725, 26ax-mp 5 . . . 4 Fun 𝑅1
28 fvelima 6909 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2927, 28mpan 689 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
3024, 29syl11 33 . 2 (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑈))
3130ssrdv 3951 1 (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3074  wss 3911  c0 4283  𝒫 cpw 4561  cima 5637  Oncon0 6318  suc csuc 6320  Fun wfun 6491   Fn wfn 6492  cfv 6497  ωcom 7803  𝑅1cr1 9699  WUnicwun 10637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-r1 9701  df-wun 10639
This theorem is referenced by:  wunom  10657
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