![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9761 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 8884 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
2 | rankon 8820 | . . . . . 6 ⊢ (rank‘𝐴) ∈ On | |
3 | omelon 8705 | . . . . . 6 ⊢ ω ∈ On | |
4 | oacl 7767 | . . . . . 6 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → ((rank‘𝐴) +𝑜 ω) ∈ On) | |
5 | 2, 3, 4 | mp2an 707 | . . . . 5 ⊢ ((rank‘𝐴) +𝑜 ω) ∈ On |
6 | peano1 7230 | . . . . . 6 ⊢ ∅ ∈ ω | |
7 | oaord1 7783 | . . . . . . 7 ⊢ (((rank‘𝐴) ∈ On ∧ ω ∈ On) → (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω))) | |
8 | 2, 3, 7 | mp2an 707 | . . . . . 6 ⊢ (∅ ∈ ω ↔ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω)) |
9 | 6, 8 | mpbi 220 | . . . . 5 ⊢ (rank‘𝐴) ∈ ((rank‘𝐴) +𝑜 ω) |
10 | r1ord2 8806 | . . . . 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 3784 | . . 3 ⊢ (𝑅1‘(rank‘𝐴)) ⊆ 𝑈 |
14 | 1, 13 | syl6ss 3761 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈) |
15 | limom 7225 | . . . . . 6 ⊢ Lim ω | |
16 | 3, 15 | pm3.2i 471 | . . . . 5 ⊢ (ω ∈ On ∧ Lim ω) |
17 | oalimcl 7792 | . . . . 5 ⊢ (((rank‘𝐴) ∈ On ∧ (ω ∈ On ∧ Lim ω)) → Lim ((rank‘𝐴) +𝑜 ω)) | |
18 | 2, 16, 17 | mp2an 707 | . . . 4 ⊢ Lim ((rank‘𝐴) +𝑜 ω) |
19 | r1limwun 9758 | . . . 4 ⊢ ((((rank‘𝐴) +𝑜 ω) ∈ On ∧ Lim ((rank‘𝐴) +𝑜 ω)) → (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni) | |
20 | 5, 18, 19 | mp2an 707 | . . 3 ⊢ (𝑅1‘((rank‘𝐴) +𝑜 ω)) ∈ WUni |
21 | 12, 20 | eqeltri 2844 | . 2 ⊢ 𝑈 ∈ WUni |
22 | 14, 21 | jctil 562 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1629 ∈ wcel 2143 ⊆ wss 3720 ∅c0 4060 Oncon0 5865 Lim wlim 5866 ‘cfv 6030 (class class class)co 6791 ωcom 7210 +𝑜 coa 7708 𝑅1cr1 8787 rankcrnk 8788 WUnicwun 9722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1868 ax-4 1883 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2145 ax-9 2152 ax-10 2172 ax-11 2188 ax-12 2201 ax-13 2406 ax-ext 2749 ax-rep 4901 ax-sep 4911 ax-nul 4919 ax-pow 4970 ax-pr 5033 ax-un 7094 ax-reg 8651 ax-inf2 8700 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1070 df-3an 1071 df-tru 1632 df-ex 1851 df-nf 1856 df-sb 2048 df-eu 2620 df-mo 2621 df-clab 2756 df-cleq 2762 df-clel 2765 df-nfc 2900 df-ne 2942 df-ral 3064 df-rex 3065 df-reu 3066 df-rab 3068 df-v 3350 df-sbc 3585 df-csb 3680 df-dif 3723 df-un 3725 df-in 3727 df-ss 3734 df-pss 3736 df-nul 4061 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4572 df-int 4609 df-iun 4653 df-br 4784 df-opab 4844 df-mpt 4861 df-tr 4884 df-id 5156 df-eprel 5161 df-po 5169 df-so 5170 df-fr 5207 df-we 5209 df-xp 5254 df-rel 5255 df-cnv 5256 df-co 5257 df-dm 5258 df-rn 5259 df-res 5260 df-ima 5261 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-oadd 7715 df-r1 8789 df-rank 8790 df-wun 9724 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |