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	⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

Proof of Theorem wunex3

Step	Hyp	Ref	Expression
1		r1rankid 8884	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (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)
5	2, 3, 4	mp2an 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)))
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 8806	. . . . 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 3784	. . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
14	1, 13	syl6ss 3761	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈)
15		limom 7225	. . . . . 6 ⊢ Lim ω
16	3, 15	pm3.2i 471	. . . . 5 ⊢ (ω ∈ On ∧ Lim ω)
17		oalimcl 7792	. . . . 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 9758	. . . 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 2844	. 2 ⊢ 𝑈 ∈ WUni
22	14, 21	jctil 562	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

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)