Theorem wunex3 10650

Description: Construct a weak universe from a given set. This version of wunex 10648 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 9769	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		rankon 9705	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
3		omelon 9553	. . . . . 6 ⊢ ω ∈ On
4		oacl 8460	. . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +o ω) ∈ On)
5	2, 3, 4	mp2an 692	. . . . 5 ⊢ ((rank‘𝐴) +o ω) ∈ On
6		peano1 7829	. . . . . 6 ⊢ ∅ ∈ ω
7		oaord1 8476	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)))
8	2, 3, 7	mp2an 692	. . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω))
9	6, 8	mpbi 230	. . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)
10		r1ord2 9691	. . . . 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 3981	. . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
14	1, 13	sstrdi 3944	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈)
15		limom 7822	. . . . . 6 ⊢ Lim ω
16	3, 15	pm3.2i 470	. . . . 5 ⊢ (ω ∈ On ∧ Lim ω)
17		oalimcl 8485	. . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +o ω))
18	2, 16, 17	mp2an 692	. . . 4 ⊢ Lim ((rank‘𝐴) +o ω)
19		r1limwun 10645	. . . 4 ⊢ ((((rank‘𝐴) +o ω) ∈ On ∧ Lim ((rank‘𝐴) +o ω)) → (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni)
20	5, 18, 19	mp2an 692	. . 3 ⊢ (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni
21	12, 20	eqeltri 2830	. 2 ⊢ 𝑈 ∈ WUni
22	14, 21	jctil 519	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3899  ∅c0 4283  Oncon0 6315  Lim wlim 6316  ‘cfv 6490  (class class class)co 7356  ωcom 7806   +_o coa 8392  𝑅₁cr1 9672  rankcrnk 9673  WUnicwun 10609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399  df-r1 9674  df-rank 9675  df-wun 10611

This theorem is referenced by: (None)