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Theorem wunex3 10784
Description: Construct a weak universe from a given set. This version of wunex 10782 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 9902 . . 3 (𝐴𝑉𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2 rankon 9838 . . . . . 6 (rank‘𝐴) ∈ On
3 omelon 9689 . . . . . 6 ω ∈ On
4 oacl 8565 . . . . . 6 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +o ω) ∈ On)
52, 3, 4mp2an 690 . . . . 5 ((rank‘𝐴) +o ω) ∈ On
6 peano1 7900 . . . . . 6 ∅ ∈ ω
7 oaord1 8581 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)))
82, 3, 7mp2an 690 . . . . . 6 (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω))
96, 8mpbi 229 . . . . 5 (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)
10 r1ord2 9824 . . . . 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, 12sseqtrri 4017 . . 3 (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
141, 13sstrdi 3992 . 2 (𝐴𝑉𝐴𝑈)
15 limom 7892 . . . . . 6 Lim ω
163, 15pm3.2i 469 . . . . 5 (ω ∈ On ∧ Lim ω)
17 oalimcl 8590 . . . . 5 (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +o ω))
182, 16, 17mp2an 690 . . . 4 Lim ((rank‘𝐴) +o ω)
19 r1limwun 10779 . . . 4 ((((rank‘𝐴) +o ω) ∈ On ∧ Lim ((rank‘𝐴) +o ω)) → (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni)
205, 18, 19mp2an 690 . . 3 (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni
2112, 20eqeltri 2822 . 2 𝑈 ∈ WUni
2214, 21jctil 518 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wss 3947  c0 4325  Oncon0 6376  Lim wlim 6377  cfv 6554  (class class class)co 7424  ωcom 7876   +o coa 8493  𝑅1cr1 9805  rankcrnk 9806  WUnicwun 10743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9635  ax-inf2 9684
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-oadd 8500  df-r1 9807  df-rank 9808  df-wun 10745
This theorem is referenced by: (None)
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