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Theorem wunr1om 9733
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 6352 . . . . . . 7 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
21eleq1d 2824 . . . . . 6 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘∅) ∈ 𝑈))
3 fveq2 6352 . . . . . . 7 (𝑥 = 𝑦 → (𝑅1𝑥) = (𝑅1𝑦))
43eleq1d 2824 . . . . . 6 (𝑥 = 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1𝑦) ∈ 𝑈))
5 fveq2 6352 . . . . . . 7 (𝑥 = suc 𝑦 → (𝑅1𝑥) = (𝑅1‘suc 𝑦))
65eleq1d 2824 . . . . . 6 (𝑥 = suc 𝑦 → ((𝑅1𝑥) ∈ 𝑈 ↔ (𝑅1‘suc 𝑦) ∈ 𝑈))
7 r10 8804 . . . . . . 7 (𝑅1‘∅) = ∅
8 wun0.1 . . . . . . . 8 (𝜑𝑈 ∈ WUni)
98wun0 9732 . . . . . . 7 (𝜑 → ∅ ∈ 𝑈)
107, 9syl5eqel 2843 . . . . . 6 (𝜑 → (𝑅1‘∅) ∈ 𝑈)
118adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝑈 ∈ WUni)
12 simpr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1𝑦) ∈ 𝑈)
1311, 12wunpw 9721 . . . . . . . 8 ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → 𝒫 (𝑅1𝑦) ∈ 𝑈)
14 nnon 7236 . . . . . . . . . 10 (𝑦 ∈ ω → 𝑦 ∈ On)
15 r1suc 8806 . . . . . . . . . 10 (𝑦 ∈ On → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1614, 15syl 17 . . . . . . . . 9 (𝑦 ∈ ω → (𝑅1‘suc 𝑦) = 𝒫 (𝑅1𝑦))
1716eleq1d 2824 . . . . . . . 8 (𝑦 ∈ ω → ((𝑅1‘suc 𝑦) ∈ 𝑈 ↔ 𝒫 (𝑅1𝑦) ∈ 𝑈))
1813, 17syl5ibr 236 . . . . . . 7 (𝑦 ∈ ω → ((𝜑 ∧ (𝑅1𝑦) ∈ 𝑈) → (𝑅1‘suc 𝑦) ∈ 𝑈))
1918expd 451 . . . . . 6 (𝑦 ∈ ω → (𝜑 → ((𝑅1𝑦) ∈ 𝑈 → (𝑅1‘suc 𝑦) ∈ 𝑈)))
202, 4, 6, 10, 19finds2 7259 . . . . 5 (𝑥 ∈ ω → (𝜑 → (𝑅1𝑥) ∈ 𝑈))
21 eleq1 2827 . . . . . 6 ((𝑅1𝑥) = 𝑦 → ((𝑅1𝑥) ∈ 𝑈𝑦𝑈))
2221imbi2d 329 . . . . 5 ((𝑅1𝑥) = 𝑦 → ((𝜑 → (𝑅1𝑥) ∈ 𝑈) ↔ (𝜑𝑦𝑈)))
2320, 22syl5ibcom 235 . . . 4 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈)))
2423rexlimiv 3165 . . 3 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦 → (𝜑𝑦𝑈))
25 r1fnon 8803 . . . . 5 𝑅1 Fn On
26 fnfun 6149 . . . . 5 (𝑅1 Fn On → Fun 𝑅1)
2725, 26ax-mp 5 . . . 4 Fun 𝑅1
28 fvelima 6410 . . . 4 ((Fun 𝑅1𝑦 ∈ (𝑅1 “ ω)) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
2927, 28mpan 708 . . 3 (𝑦 ∈ (𝑅1 “ ω) → ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑦)
3024, 29syl11 33 . 2 (𝜑 → (𝑦 ∈ (𝑅1 “ ω) → 𝑦𝑈))
3130ssrdv 3750 1 (𝜑 → (𝑅1 “ ω) ⊆ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051  wss 3715  c0 4058  𝒫 cpw 4302  cima 5269  Oncon0 5884  suc csuc 5886  Fun wfun 6043   Fn wfn 6044  cfv 6049  ωcom 7230  𝑅1cr1 8798  WUnicwun 9714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-wun 9716
This theorem is referenced by:  wunom  9734
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