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Mirrors > Home > MPE Home > Th. List > wunex3 | Structured version Visualization version GIF version |
Description: Construct a weak universe from a given set. This version of wunex 10782 has a simpler proof, but requires the axiom of regularity. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wunex3.u | ⊢ 𝑈 = (𝑅1‘((rank‘𝐴) +o ω)) |
Ref | Expression |
---|---|
wunex3 | ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1rankid 9902 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
2 | rankon 9838 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | omelon 9689 | . . . . . 6 ⊢ ω ∈ On | |
4 | oacl 8565 | . . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +o ω) ∈ On) | |
5 | 2, 3, 4 | mp2an 690 | . . . . 5 ⊢ ((rank‘𝐴) +o ω) ∈ On |
6 | peano1 7900 | . . . . . 6 ⊢ ∅ ∈ ω | |
7 | oaord1 8581 | . . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω))) | |
8 | 2, 3, 7 | mp2an 690 | . . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω)) |
9 | 6, 8 | mpbi 229 | . . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω) |
10 | r1ord2 9824 | . . . . 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 4017 | . . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈 |
14 | 1, 13 | sstrdi 3992 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈) |
15 | limom 7892 | . . . . . 6 ⊢ Lim ω | |
16 | 3, 15 | pm3.2i 469 | . . . . 5 ⊢ (ω ∈ On ∧ Lim ω) |
17 | oalimcl 8590 | . . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +o ω)) | |
18 | 2, 16, 17 | mp2an 690 | . . . 4 ⊢ Lim ((rank‘𝐴) +o ω) |
19 | r1limwun 10779 | . . . 4 ⊢ ((((rank‘𝐴) +o ω) ∈ On ∧ Lim ((rank‘𝐴) +o ω)) → (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni) | |
20 | 5, 18, 19 | mp2an 690 | . . 3 ⊢ (𝑅1‘((rank‘𝐴) +o ω)) ∈ WUni |
21 | 12, 20 | eqeltri 2822 | . 2 ⊢ 𝑈 ∈ WUni |
22 | 14, 21 | jctil 518 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ∅c0 4325 Oncon0 6376 Lim wlim 6377 ‘cfv 6554 (class class class)co 7424 ωcom 7876 +o coa 8493 𝑅1cr1 9805 rankcrnk 9806 WUnicwun 10743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-reg 9635 ax-inf2 9684 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-oadd 8500 df-r1 9807 df-rank 9808 df-wun 10745 |
This theorem is referenced by: (None) |
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