Theorem wunex3 9563

Description: Construct a weak universe from a given set. This version of wunex 9561 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 8722	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		rankon 8658	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
3		omelon 8543	. . . . . 6 ⊢ ω ∈ On
4		oacl 7615	. . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +𝑜 ω) ∈ On)
5	2, 3, 4	mp2an 708	. . . . 5 ⊢ ((rank‘𝐴) +𝑜 ω) ∈ On
6		peano1 7085	. . . . . 6 ⊢ ∅ ∈ ω
7		oaord1 7631	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)))
8	2, 3, 7	mp2an 708	. . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω))
9	6, 8	mpbi 220	. . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)
10		r1ord2 8644	. . . . 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 3638	. . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
14	1, 13	syl6ss 3615	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈)
15		limom 7080	. . . . . 6 ⊢ Lim ω
16	3, 15	pm3.2i 471	. . . . 5 ⊢ (ω ∈ On ∧ Lim ω)
17		oalimcl 7640	. . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +𝑜 ω))
18	2, 16, 17	mp2an 708	. . . 4 ⊢ Lim ((rank‘𝐴) +𝑜 ω)
19		r1limwun 9558	. . . 4 ⊢ ((((rank‘𝐴) +𝑜 ω) ∈ On ∧ Lim ((rank‘𝐴) +𝑜 ω)) → (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni)
20	5, 18, 19	mp2an 708	. . 3 ⊢ (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni
21	12, 20	eqeltri 2697	. 2 ⊢ 𝑈 ∈ WUni
22	14, 21	jctil 560	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∅c0 3915  Oncon0 5723  Lim wlim 5724  ‘cfv 5888  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557  𝑅₁cr1 8625  rankcrnk 8626  WUnicwun 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-r1 8627  df-rank 8628  df-wun 9524

This theorem is referenced by: (None)