Theorem wunex3 10721

Description: Construct a weak universe from a given set. This version of wunex 10719 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 9827	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		rankon 9763	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
3		omelon 9611	. . . . . 6 ⊢ ω ∈ On
4		oacl 8516	. . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +o ω) ∈ On)
5	2, 3, 4	mp2an 704	. . . . 5 ⊢ ((rank‘𝐴) +o ω) ∈ On
6		peano1 7881	. . . . . 6 ⊢ ∅ ∈ ω
7		oaord1 8532	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)))
8	2, 3, 7	mp2an 704	. . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω))
9	6, 8	mpbi 233	. . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)
10		r1ord2 9749	. . . . 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	sseqtrri 3986	. . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
14	1, 13	sstrdi 3949	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈)
15		limom 7874	. . . . . 6 ⊢ Lim ω
16	3, 15	pm3.2i 475	. . . . 5 ⊢ (ω ∈ On ∧ Lim ω)
17		oalimcl 8541	. . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +o ω))
18	2, 16, 17	mp2an 704	. . . 4 ⊢ Lim ((rank‘𝐴) +o ω)
19		r1limwun 10716	. . . 4 ⊢ ((((rank‘𝐴) +o ω) ∈ On ∧ Lim ((rank‘𝐴) +o ω)) → (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni)
20	5, 18, 19	mp2an 704	. . 3 ⊢ (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni
21	12, 20	eqeltri 2859	. 2 ⊢ 𝑈 ∈ WUni
22	14, 21	jctil 528	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  ∅c0 4286  Oncon0 6360  Lim wlim 6361  ‘cfv 6536  (class class class)co 7410  ωcom 7858   +_o coa 8446  𝑅₁cr1 9730  rankcrnk 9731  WUnicwun 10680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-oadd 8453  df-r1 9732  df-rank 9733  df-wun 10682

This theorem is referenced by: (None)