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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 10777 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 9897 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
2 | rankon 9833 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | omelon 9684 | . . . . . 6 ⊢ ω ∈ On | |
4 | oacl 8572 | . . . . . 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 7911 | . . . . . 6 ⊢ ∅ ∈ ω | |
7 | oaord1 8588 | . . . . . . 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 9819 | . . . . 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 4008 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈) |
15 | limom 7903 | . . . . . 6 ⊢ Lim ω | |
16 | 3, 15 | pm3.2i 470 | . . . . 5 ⊢ (ω ∈ On ∧ Lim ω) |
17 | oalimcl 8597 | . . . . 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 10774 | . . . 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 2835 | . 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 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 Oncon0 6386 Lim wlim 6387 ‘cfv 6563 (class class class)co 7431 ωcom 7887 +o coa 8502 𝑅1cr1 9800 rankcrnk 9801 WUnicwun 10738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 df-r1 9802 df-rank 9803 df-wun 10740 |
This theorem is referenced by: (None) |
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