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Theorem wunex3 9507
Description: Construct a weak universe from a given set. This version of wunex 9505 has a simpler proof, but requires the axiom of regularity. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypothesis
Ref Expression
wunex3.u 𝑈 = (𝑅1‘((rank‘𝐴) +𝑜 ω))
Assertion
Ref Expression
wunex3 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))

Proof of Theorem wunex3
StepHypRef Expression
1 r1rankid 8666 . . 3 (𝐴𝑉𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2 rankon 8602 . . . . . 6 (rank‘𝐴) ∈ On
3 omelon 8487 . . . . . 6 ω ∈ On
4 oacl 7560 . . . . . 6 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +𝑜 ω) ∈ On)
52, 3, 4mp2an 707 . . . . 5 ((rank‘𝐴) +𝑜 ω) ∈ On
6 peano1 7032 . . . . . 6 ∅ ∈ ω
7 oaord1 7576 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)))
82, 3, 7mp2an 707 . . . . . 6 (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω))
96, 8mpbi 220 . . . . 5 (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)
10 r1ord2 8588 . . . . 5 (((rank‘𝐴) +𝑜 ω) ∈ On → ((rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω) → (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) +𝑜 ω))))
115, 9, 10mp2 9 . . . 4 (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) +𝑜 ω))
12 wunex3.u . . . 4 𝑈 = (𝑅1‘((rank‘𝐴) +𝑜 ω))
1311, 12sseqtr4i 3617 . . 3 (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
141, 13syl6ss 3595 . 2 (𝐴𝑉𝐴𝑈)
15 limom 7027 . . . . . 6 Lim ω
163, 15pm3.2i 471 . . . . 5 (ω ∈ On ∧ Lim ω)
17 oalimcl 7585 . . . . 5 (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +𝑜 ω))
182, 16, 17mp2an 707 . . . 4 Lim ((rank‘𝐴) +𝑜 ω)
19 r1limwun 9502 . . . 4 ((((rank‘𝐴) +𝑜 ω) ∈ On ∧ Lim ((rank‘𝐴) +𝑜 ω)) → (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni)
205, 18, 19mp2an 707 . . 3 (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni
2112, 20eqeltri 2694 . 2 𝑈 ∈ WUni
2214, 21jctil 559 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3555  c0 3891  Oncon0 5682  Lim wlim 5683  cfv 5847  (class class class)co 6604  ωcom 7012   +𝑜 coa 7502  𝑅1cr1 8569  rankcrnk 8570  WUnicwun 9466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-r1 8571  df-rank 8572  df-wun 9468
This theorem is referenced by: (None)
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