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Theorem wunex3 9763
Description: Construct a weak universe from a given set. This version of wunex 9761 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 8884 . . 3 (𝐴𝑉𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2 rankon 8820 . . . . . 6 (rank‘𝐴) ∈ On
3 omelon 8705 . . . . . 6 ω ∈ On
4 oacl 7767 . . . . . 6 (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +𝑜 ω) ∈ On)
52, 3, 4mp2an 707 . . . . 5 ((rank‘𝐴) +𝑜 ω) ∈ On
6 peano1 7230 . . . . . 6 ∅ ∈ ω
7 oaord1 7783 . . . . . . 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 8806 . . . . 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 3784 . . 3 (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
141, 13syl6ss 3761 . 2 (𝐴𝑉𝐴𝑈)
15 limom 7225 . . . . . 6 Lim ω
163, 15pm3.2i 471 . . . . 5 (ω ∈ On ∧ Lim ω)
17 oalimcl 7792 . . . . 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 9758 . . . 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 2844 . 2 𝑈 ∈ WUni
2214, 21jctil 562 1 (𝐴𝑉 → (𝑈 ∈ WUni ∧ 𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1629  wcel 2143  wss 3720  c0 4060  Oncon0 5865  Lim wlim 5866  cfv 6030  (class class class)co 6791  ωcom 7210   +𝑜 coa 7708  𝑅1cr1 8787  rankcrnk 8788  WUnicwun 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-reg 8651  ax-inf2 8700
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-int 4609  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-oadd 7715  df-r1 8789  df-rank 8790  df-wun 9724
This theorem is referenced by: (None)
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