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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 10163 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 9290 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
2 | rankon 9226 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | omelon 9111 | . . . . . 6 ⊢ ω ∈ On | |
4 | oacl 8162 | . . . . . 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 7603 | . . . . . 6 ⊢ ∅ ∈ ω | |
7 | oaord1 8179 | . . . . . . 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 232 | . . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +o ω) |
10 | r1ord2 9212 | . . . . 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 4006 | . . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈 |
14 | 1, 13 | sstrdi 3981 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈) |
15 | limom 7597 | . . . . . 6 ⊢ Lim ω | |
16 | 3, 15 | pm3.2i 473 | . . . . 5 ⊢ (ω ∈ On ∧ Lim ω) |
17 | oalimcl 8188 | . . . . 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 10160 | . . . 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 2911 | . 2 ⊢ 𝑈 ∈ WUni |
22 | 14, 21 | jctil 522 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∅c0 4293 Oncon0 6193 Lim wlim 6194 ‘cfv 6357 (class class class)co 7158 ωcom 7582 +o coa 8101 𝑅1cr1 9193 rankcrnk 9194 WUnicwun 10124 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-reg 9058 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-oadd 8108 df-r1 9195 df-rank 9196 df-wun 10126 |
This theorem is referenced by: (None) |
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