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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 9440 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 8605 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
2 | rankon 8541 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | omelon 8426 | . . . . . 6 ⊢ ω ∈ On | |
4 | oacl 7502 | . . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +𝑜 ω) ∈ On) | |
5 | 2, 3, 4 | mp2an 704 | . . . . 5 ⊢ ((rank‘𝐴) +𝑜 ω) ∈ On |
6 | peano1 6977 | . . . . . 6 ⊢ ∅ ∈ ω | |
7 | oaord1 7518 | . . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω))) | |
8 | 2, 3, 7 | mp2an 704 | . . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)) |
9 | 6, 8 | mpbi 219 | . . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω) |
10 | r1ord2 8527 | . . . . 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 | sseqtr4i 3601 | . . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈 |
14 | 1, 13 | syl6ss 3580 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈) |
15 | limom 6972 | . . . . . 6 ⊢ Lim ω | |
16 | 3, 15 | pm3.2i 470 | . . . . 5 ⊢ (ω ∈ On ∧ Lim ω) |
17 | oalimcl 7527 | . . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +𝑜 ω)) | |
18 | 2, 16, 17 | mp2an 704 | . . . 4 ⊢ Lim ((rank‘𝐴) +𝑜 ω) |
19 | r1limwun 9437 | . . . 4 ⊢ ((((rank‘𝐴) +𝑜 ω) ∈ On ∧ Lim ((rank‘𝐴) +𝑜 ω)) → (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni) | |
20 | 5, 18, 19 | mp2an 704 | . . 3 ⊢ (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni |
21 | 12, 20 | eqeltri 2684 | . 2 ⊢ 𝑈 ∈ WUni |
22 | 14, 21 | jctil 558 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 Oncon0 5640 Lim wlim 5641 ‘cfv 5804 (class class class)co 6549 ωcom 6957 +𝑜 coa 7444 𝑅1cr1 8508 rankcrnk 8509 WUnicwun 9401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-r1 8510 df-rank 8511 df-wun 9403 |
This theorem is referenced by: (None) |
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