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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 11155 | . 2 ⊢ ℂ = (R × R) | |
2 | wuncn.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | df-nr 11090 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
4 | df-ni 10906 | . . . . . . . . . . . 12 ⊢ N = (ω ∖ {∅}) | |
5 | wuncn.2 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | 2, 5 | wundif 10748 | . . . . . . . . . . . 12 ⊢ (𝜑 → (ω ∖ {∅}) ∈ 𝑈) |
7 | 4, 6 | eqeltrid 2830 | . . . . . . . . . . 11 ⊢ (𝜑 → N ∈ 𝑈) |
8 | 2, 7, 7 | wunxp 10758 | . . . . . . . . . 10 ⊢ (𝜑 → (N × N) ∈ 𝑈) |
9 | elpqn 10959 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
10 | 9 | ssriv 3982 | . . . . . . . . . . 11 ⊢ Q ⊆ (N × N) |
11 | 10 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → Q ⊆ (N × N)) |
12 | 2, 8, 11 | wunss 10746 | . . . . . . . . 9 ⊢ (𝜑 → Q ∈ 𝑈) |
13 | 2, 12 | wunpw 10741 | . . . . . . . 8 ⊢ (𝜑 → 𝒫 Q ∈ 𝑈) |
14 | prpssnq 11024 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → 𝑥 ⊊ Q) | |
15 | 14 | pssssd 4093 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ P → 𝑥 ⊆ Q) |
16 | velpw 4602 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 Q ↔ 𝑥 ⊆ Q) | |
17 | 15, 16 | sylibr 233 | . . . . . . . . . 10 ⊢ (𝑥 ∈ P → 𝑥 ∈ 𝒫 Q) |
18 | 17 | ssriv 3982 | . . . . . . . . 9 ⊢ P ⊆ 𝒫 Q |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → P ⊆ 𝒫 Q) |
20 | 2, 13, 19 | wunss 10746 | . . . . . . 7 ⊢ (𝜑 → P ∈ 𝑈) |
21 | 2, 20, 20 | wunxp 10758 | . . . . . 6 ⊢ (𝜑 → (P × P) ∈ 𝑈) |
22 | 2, 21 | wunpw 10741 | . . . . 5 ⊢ (𝜑 → 𝒫 (P × P) ∈ 𝑈) |
23 | enrer 11097 | . . . . . . 7 ⊢ ~R Er (P × P) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ~R Er (P × P)) |
25 | 24 | qsss 8799 | . . . . 5 ⊢ (𝜑 → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
26 | 2, 22, 25 | wunss 10746 | . . . 4 ⊢ (𝜑 → ((P × P) / ~R ) ∈ 𝑈) |
27 | 3, 26 | eqeltrid 2830 | . . 3 ⊢ (𝜑 → R ∈ 𝑈) |
28 | 2, 27, 27 | wunxp 10758 | . 2 ⊢ (𝜑 → (R × R) ∈ 𝑈) |
29 | 1, 28 | eqeltrid 2830 | 1 ⊢ (𝜑 → ℂ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∖ cdif 3943 ⊆ wss 3946 ∅c0 4322 𝒫 cpw 4597 {csn 4623 × cxp 5672 ωcom 7868 Er wer 8723 / cqs 8725 WUnicwun 10734 Ncnpi 10878 Qcnq 10886 Pcnp 10893 ~R cer 10898 Rcnr 10899 ℂcc 11147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-oadd 8492 df-omul 8493 df-er 8726 df-ec 8728 df-qs 8732 df-wun 10736 df-ni 10906 df-pli 10907 df-mi 10908 df-lti 10909 df-plpq 10942 df-mpq 10943 df-ltpq 10944 df-enq 10945 df-nq 10946 df-erq 10947 df-plq 10948 df-mq 10949 df-1nq 10950 df-rq 10951 df-ltnq 10952 df-np 11015 df-plp 11017 df-ltp 11019 df-enr 11089 df-nr 11090 df-c 11155 |
This theorem is referenced by: wunndx 17192 |
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