Theorem wunex3 10826

Description: Construct a weak universe from a given set. This version of wunex 10824 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 9875	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2		rankon 9803	. . . . . 6 ⊢ (rank‘𝐴) ∈ On
3		omelon 9647	. . . . . 6 ⊢ ω ∈ On
4		oacl 8543	. . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +o ω) ∈ On)
5	2, 3, 4	mp2an 705	. . . . 5 ⊢ ((rank‘𝐴) +o ω) ∈ On
6		peano1 7900	. . . . . 6 ⊢ ∅ ∈ ω
7		oaord1 8559	. . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)))
8	2, 3, 7	mp2an 705	. . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω))
9	6, 8	mpbi 233	. . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)
10		r1ord2 9788	. . . . 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 3980	. . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈
14	1, 13	sstrdi 3943	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈)
15		limom 7893	. . . . . 6 ⊢ Lim ω
16	3, 15	pm3.2i 476	. . . . 5 ⊢ (ω ∈ On ∧ Lim ω)
17		oalimcl 8568	. . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +o ω))
18	2, 16, 17	mp2an 705	. . . 4 ⊢ Lim ((rank‘𝐴) +o ω)
19		r1limwun 10821	. . . 4 ⊢ ((((rank‘𝐴) +o ω) ∈ On ∧ Lim ((rank‘𝐴) +o ω)) → (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni)
20	5, 18, 19	mp2an 705	. . 3 ⊢ (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni
21	12, 20	eqeltri 2857	. 2 ⊢ 𝑈 ∈ WUni
22	14, 21	jctil 529	1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ∅c0 4279  Oncon0 6362  Lim wlim 6363  ‘cfv 6538  (class class class)co 7420  ωcom 7877   +_o coa 8473  𝑅₁cr1 9766  rankcrnk 9767  WUnicwun 10785

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-oadd 8480  df-r1 9768  df-rank 9769  df-wun 10787

This theorem is used by: (None)