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Mirrors > Home > MPE Home > Th. List > wuncn | Structured version Visualization version GIF version |
Description: A weak universe containing ω contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wuncn.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wuncn.2 | ⊢ (𝜑 → ω ∈ 𝑈) |
Ref | Expression |
---|---|
wuncn | ⊢ (𝜑 → ℂ ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 11115 | . 2 ⊢ ℂ = (R × R) | |
2 | wuncn.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | df-nr 11050 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
4 | df-ni 10866 | . . . . . . . . . . . 12 ⊢ N = (ω ∖ {∅}) | |
5 | wuncn.2 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | 2, 5 | wundif 10708 | . . . . . . . . . . . 12 ⊢ (𝜑 → (ω ∖ {∅}) ∈ 𝑈) |
7 | 4, 6 | eqeltrid 2837 | . . . . . . . . . . 11 ⊢ (𝜑 → N ∈ 𝑈) |
8 | 2, 7, 7 | wunxp 10718 | . . . . . . . . . 10 ⊢ (𝜑 → (N × N) ∈ 𝑈) |
9 | elpqn 10919 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
10 | 9 | ssriv 3986 | . . . . . . . . . . 11 ⊢ Q ⊆ (N × N) |
11 | 10 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → Q ⊆ (N × N)) |
12 | 2, 8, 11 | wunss 10706 | . . . . . . . . 9 ⊢ (𝜑 → Q ∈ 𝑈) |
13 | 2, 12 | wunpw 10701 | . . . . . . . 8 ⊢ (𝜑 → 𝒫 Q ∈ 𝑈) |
14 | prpssnq 10984 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → 𝑥 ⊊ Q) | |
15 | 14 | pssssd 4097 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ P → 𝑥 ⊆ Q) |
16 | velpw 4607 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 Q ↔ 𝑥 ⊆ Q) | |
17 | 15, 16 | sylibr 233 | . . . . . . . . . 10 ⊢ (𝑥 ∈ P → 𝑥 ∈ 𝒫 Q) |
18 | 17 | ssriv 3986 | . . . . . . . . 9 ⊢ P ⊆ 𝒫 Q |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → P ⊆ 𝒫 Q) |
20 | 2, 13, 19 | wunss 10706 | . . . . . . 7 ⊢ (𝜑 → P ∈ 𝑈) |
21 | 2, 20, 20 | wunxp 10718 | . . . . . 6 ⊢ (𝜑 → (P × P) ∈ 𝑈) |
22 | 2, 21 | wunpw 10701 | . . . . 5 ⊢ (𝜑 → 𝒫 (P × P) ∈ 𝑈) |
23 | enrer 11057 | . . . . . . 7 ⊢ ~R Er (P × P) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ~R Er (P × P)) |
25 | 24 | qsss 8771 | . . . . 5 ⊢ (𝜑 → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
26 | 2, 22, 25 | wunss 10706 | . . . 4 ⊢ (𝜑 → ((P × P) / ~R ) ∈ 𝑈) |
27 | 3, 26 | eqeltrid 2837 | . . 3 ⊢ (𝜑 → R ∈ 𝑈) |
28 | 2, 27, 27 | wunxp 10718 | . 2 ⊢ (𝜑 → (R × R) ∈ 𝑈) |
29 | 1, 28 | eqeltrid 2837 | 1 ⊢ (𝜑 → ℂ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 × cxp 5674 ωcom 7854 Er wer 8699 / cqs 8701 WUnicwun 10694 Ncnpi 10838 Qcnq 10846 Pcnp 10853 ~R cer 10858 Rcnr 10859 ℂcc 11107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-omul 8470 df-er 8702 df-ec 8704 df-qs 8708 df-wun 10696 df-ni 10866 df-pli 10867 df-mi 10868 df-lti 10869 df-plpq 10902 df-mpq 10903 df-ltpq 10904 df-enq 10905 df-nq 10906 df-erq 10907 df-plq 10908 df-mq 10909 df-1nq 10910 df-rq 10911 df-ltnq 10912 df-np 10975 df-plp 10977 df-ltp 10979 df-enr 11049 df-nr 11050 df-c 11115 |
This theorem is referenced by: wunndx 17127 |
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