Theorem wunex3 9548

Description: Construct a weak universe from a given set. This version of wunex 9546 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	⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

Proof of Theorem wunex3

Step	Hyp	Ref	Expression
1		r1rankid 8707	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		rankon 8643	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
3		omelon 8528	. . . . . 6 ⊢ ω ∈ On
4		oacl 7600	. . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +𝑜 ω) ∈ On)
5	2, 3, 4	mp2an 707	. . . . 5 ⊢ ((rank‘𝐴) +𝑜 ω) ∈ On
6		peano1 7070	. . . . . 6 ⊢ ∅ ∈ ω
7		oaord1 7616	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)))
8	2, 3, 7	mp2an 707	. . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω))
9	6, 8	mpbi 220	. . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)
10		r1ord2 8629	. . . . 5 ⊢ (((rank‘𝐴) +𝑜 ω) ∈ On → ((rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω) → (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) +𝑜 ω))))
11	5, 9, 10	mp2 9	. . . 4 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ (𝑅1‘((rank‘𝐴) +𝑜 ω))
12		wunex3.u	. . . 4 ⊢ 𝑈 = (𝑅1‘((rank‘𝐴) +𝑜 ω))
13	11, 12	sseqtr4i 3630	. . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
14	1, 13	syl6ss 3607	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈)
15		limom 7065	. . . . . 6 ⊢ Lim ω
16	3, 15	pm3.2i 471	. . . . 5 ⊢ (ω ∈ On ∧ Lim ω)
17		oalimcl 7625	. . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +𝑜 ω))
18	2, 16, 17	mp2an 707	. . . 4 ⊢ Lim ((rank‘𝐴) +𝑜 ω)
19		r1limwun 9543	. . . 4 ⊢ ((((rank‘𝐴) +𝑜 ω) ∈ On ∧ Lim ((rank‘𝐴) +𝑜 ω)) → (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni)
20	5, 18, 19	mp2an 707	. . 3 ⊢ (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni
21	12, 20	eqeltri 2695	. 2 ⊢ 𝑈 ∈ WUni
22	14, 21	jctil 559	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988   ⊆ wss 3567  ∅c0 3907  Oncon0 5711  Lim wlim 5712  ‘cfv 5876  (class class class)co 6635  ωcom 7050   +_𝑜 coa 7542  𝑅₁cr1 8610  rankcrnk 8611  WUnicwun 9507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482  ax-inf2 8523

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-r1 8612  df-rank 8613  df-wun 9509

This theorem is referenced by: (None)