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Theorem wunr1om 9420
 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 r1fnon 8513 . . . . . 6 𝑅1 Fn On
2 fnfun 5902 . . . . . 6 (𝑅1 Fn On → Fun 𝑅1)
31, 2ax-mp 5 . . . . 5 Fun 𝑅1
4 fvelima 6158 . . . . 5 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
53, 4mpan 702 . . . 4 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
6 fveq2 6103 . . . . . . . 8 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
76eleq1d 2672 . . . . . . 7 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
8 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
98eleq1d 2672 . . . . . . 7 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1𝑦) ∈ 𝑈))
10 fveq2 6103 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
1110eleq1d 2672 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
12 r10 8514 . . . . . . . 8 (𝑅1‘∅) = ∅
13 wun0.1 . . . . . . . . 9 (𝜑𝑈 ∈ WUni)
1413wun0 9419 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑈)
1512, 14syl5eqel 2692 . . . . . . 7 (𝜑 → (𝑅1‘∅) ∈ 𝑈)
1613adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝑈 ∈ WUni)
17 simpr 476 . . . . . . . . . 10 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1𝑦) ∈ 𝑈)
1816, 17wunpw 9408 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝒫 (𝑅1𝑦) ∈ 𝑈)
19 nnon 6963 . . . . . . . . . . 11 (𝑦 ∈ ω → 𝑦 ∈ On)
20 r1suc 8516 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
2119, 20syl 17 . . . . . . . . . 10 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
2221eleq1d 2672 . . . . . . . . 9 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑈))
2318, 22syl5ibr 235 . . . . . . . 8 (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈))
2423expd 451 . . . . . . 7 (𝑦 ∈ ω → (𝜑 → ((𝑅1𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈)))
257, 9, 11, 15, 24finds2 6986 . . . . . 6 (𝑥 ∈ ω → (𝜑 → (𝑅1𝑥) ∈ 𝑈))
26 eleq1 2676 . . . . . . 7 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑈𝑦𝑈))
2726imbi2d 329 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝜑 → (𝑅1𝑥) ∈ 𝑈) ↔ (𝜑𝑦𝑈)))
2825, 27syl5ibcom 234 . . . . 5 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈)))
2928rexlimiv 3009 . . . 4 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈))
305, 29syl 17 . . 3 (𝑦 ∈ (𝑅1 “ ω) → (𝜑𝑦𝑈))
3130com12 32 . 2 (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑈))
3231ssrdv 3574 1 (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   “ cima 5041  Oncon0 5640  suc csuc 5642  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  ωcom 6957  𝑅1cr1 8508  WUnicwun 9401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-wun 9403 This theorem is referenced by:  wunom  9421
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