Theorem wunex3 9848

Description: Construct a weak universe from a given set. This version of wunex 9846 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 8969	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		rankon 8905	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
3		omelon 8790	. . . . . 6 ⊢ ω ∈ On
4		oacl 7852	. . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +𝑜 ω) ∈ On)
5	2, 3, 4	mp2an 675	. . . . 5 ⊢ ((rank‘𝐴) +𝑜 ω) ∈ On
6		peano1 7315	. . . . . 6 ⊢ ∅ ∈ ω
7		oaord1 7868	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)))
8	2, 3, 7	mp2an 675	. . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω))
9	6, 8	mpbi 221	. . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)
10		r1ord2 8891	. . . . 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 3835	. . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
14	1, 13	syl6ss 3810	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈)
15		limom 7310	. . . . . 6 ⊢ Lim ω
16	3, 15	pm3.2i 458	. . . . 5 ⊢ (ω ∈ On ∧ Lim ω)
17		oalimcl 7877	. . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +𝑜 ω))
18	2, 16, 17	mp2an 675	. . . 4 ⊢ Lim ((rank‘𝐴) +𝑜 ω)
19		r1limwun 9843	. . . 4 ⊢ ((((rank‘𝐴) +𝑜 ω) ∈ On ∧ Lim ((rank‘𝐴) +𝑜 ω)) → (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni)
20	5, 18, 19	mp2an 675	. . 3 ⊢ (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni
21	12, 20	eqeltri 2881	. 2 ⊢ 𝑈 ∈ WUni
22	14, 21	jctil 511	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1637   ∈ wcel 2156   ⊆ wss 3769  ∅c0 4116  Oncon0 5936  Lim wlim 5937  ‘cfv 6101  (class class class)co 6874  ωcom 7295   +_𝑜 coa 7793  𝑅₁cr1 8872  rankcrnk 8873  WUnicwun 9807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-reg 8736  ax-inf2 8785

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-oadd 7800  df-r1 8874  df-rank 8875  df-wun 9809

This theorem is referenced by: (None)