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Theorem wunex3 10781
Description: Construct a weak universe from a given set. This version of wunex 10779 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 9899 . . 3 (𝐴𝑉𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2 rankon 9835 . . . . . 6 (rank‘𝐴) ∈ On
3 omelon 9686 . . . . . 6 ω ∈ On
4 oacl 8573 . . . . . 6 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +o ω) ∈ On)
52, 3, 4mp2an 692 . . . . 5 ((rank‘𝐴) +o ω) ∈ On
6 peano1 7910 . . . . . 6 ∅ ∈ ω
7 oaord1 8589 . . . . . . 7 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)))
82, 3, 7mp2an 692 . . . . . 6 (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω))
96, 8mpbi 230 . . . . 5 (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)
10 r1ord2 9821 . . . . 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 4033 . . 3 (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
141, 13sstrdi 3996 . 2 (𝐴𝑉𝐴𝑈)
15 limom 7903 . . . . . 6 Lim ω
163, 15pm3.2i 470 . . . . 5 (ω ∈ On ∧ Lim ω)
17 oalimcl 8598 . . . . 5 (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +o ω))
182, 16, 17mp2an 692 . . . 4 Lim ((rank‘𝐴) +o ω)
19 r1limwun 10776 . . . 4 ((((rank‘𝐴) +o ω) ∈ On ∧ Lim ((rank‘𝐴) +o ω)) → (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni)
205, 18, 19mp2an 692 . . 3 (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni
2112, 20eqeltri 2837 . 2 𝑈 ∈ WUni
2214, 21jctil 519 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  c0 4333  Oncon0 6384  Lim wlim 6385  cfv 6561  (class class class)co 7431  ωcom 7887   +o coa 8503  𝑅1cr1 9802  rankcrnk 9803  WUnicwun 10740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-r1 9804  df-rank 9805  df-wun 10742
This theorem is referenced by: (None)
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