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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 9821 | . 2 ⊢ ℂ = (R × R) | |
2 | wuncn.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | df-nr 9757 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
4 | df-ni 9573 | . . . . . . . . . . . 12 ⊢ N = (ω ∖ {∅}) | |
5 | wuncn.2 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | 2, 5 | wundif 9415 | . . . . . . . . . . . 12 ⊢ (𝜑 → (ω ∖ {∅}) ∈ 𝑈) |
7 | 4, 6 | syl5eqel 2692 | . . . . . . . . . . 11 ⊢ (𝜑 → N ∈ 𝑈) |
8 | 2, 7, 7 | wunxp 9425 | . . . . . . . . . 10 ⊢ (𝜑 → (N × N) ∈ 𝑈) |
9 | elpqn 9626 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
10 | 9 | ssriv 3572 | . . . . . . . . . . 11 ⊢ Q ⊆ (N × N) |
11 | 10 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → Q ⊆ (N × N)) |
12 | 2, 8, 11 | wunss 9413 | . . . . . . . . 9 ⊢ (𝜑 → Q ∈ 𝑈) |
13 | 2, 12 | wunpw 9408 | . . . . . . . 8 ⊢ (𝜑 → 𝒫 Q ∈ 𝑈) |
14 | prpssnq 9691 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → 𝑥 ⊊ Q) | |
15 | 14 | pssssd 3666 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ P → 𝑥 ⊆ Q) |
16 | selpw 4115 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 Q ↔ 𝑥 ⊆ Q) | |
17 | 15, 16 | sylibr 223 | . . . . . . . . . 10 ⊢ (𝑥 ∈ P → 𝑥 ∈ 𝒫 Q) |
18 | 17 | ssriv 3572 | . . . . . . . . 9 ⊢ P ⊆ 𝒫 Q |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → P ⊆ 𝒫 Q) |
20 | 2, 13, 19 | wunss 9413 | . . . . . . 7 ⊢ (𝜑 → P ∈ 𝑈) |
21 | 2, 20, 20 | wunxp 9425 | . . . . . 6 ⊢ (𝜑 → (P × P) ∈ 𝑈) |
22 | 2, 21 | wunpw 9408 | . . . . 5 ⊢ (𝜑 → 𝒫 (P × P) ∈ 𝑈) |
23 | enrer 9765 | . . . . . . 7 ⊢ ~R Er (P × P) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ~R Er (P × P)) |
25 | 24 | qsss 7695 | . . . . 5 ⊢ (𝜑 → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
26 | 2, 22, 25 | wunss 9413 | . . . 4 ⊢ (𝜑 → ((P × P) / ~R ) ∈ 𝑈) |
27 | 3, 26 | syl5eqel 2692 | . . 3 ⊢ (𝜑 → R ∈ 𝑈) |
28 | 2, 27, 27 | wunxp 9425 | . 2 ⊢ (𝜑 → (R × R) ∈ 𝑈) |
29 | 1, 28 | syl5eqel 2692 | 1 ⊢ (𝜑 → ℂ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 × cxp 5036 ωcom 6957 Er wer 7626 / cqs 7628 WUnicwun 9401 Ncnpi 9545 Qcnq 9553 Pcnp 9560 ~R cer 9565 Rcnr 9566 ℂcc 9813 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 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-rmo 2904 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-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-wun 9403 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 df-np 9682 df-plp 9684 df-ltp 9686 df-enr 9756 df-nr 9757 df-c 9821 |
This theorem is referenced by: wunndx 15711 |
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