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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 10779 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 9899 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
| 2 | rankon 9835 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
| 3 | omelon 9686 | . . . . . 6 ⊢ ω ∈ On | |
| 4 | oacl 8573 | . . . . . 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 7910 | . . . . . 6 ⊢ ∅ ∈ ω | |
| 7 | oaord1 8589 | . . . . . . 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 9821 | . . . . 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 4033 | . . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈 |
| 14 | 1, 13 | sstrdi 3996 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈) |
| 15 | limom 7903 | . . . . . 6 ⊢ Lim ω | |
| 16 | 3, 15 | pm3.2i 470 | . . . . 5 ⊢ (ω ∈ On ∧ Lim ω) |
| 17 | oalimcl 8598 | . . . . 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 10776 | . . . 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 2837 | . 2 ⊢ 𝑈 ∈ WUni |
| 22 | 14, 21 | jctil 519 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 Oncon0 6384 Lim wlim 6385 ‘cfv 6561 (class class class)co 7431 ωcom 7887 +o coa 8503 𝑅1cr1 9802 rankcrnk 9803 WUnicwun 10740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-oadd 8510 df-r1 9804 df-rank 9805 df-wun 10742 |
| This theorem is referenced by: (None) |
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