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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 11064 | . 2 ⊢ ℂ = (R × R) | |
2 | wuncn.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | df-nr 10999 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
4 | df-ni 10815 | . . . . . . . . . . . 12 ⊢ N = (ω ∖ {∅}) | |
5 | wuncn.2 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | 2, 5 | wundif 10657 | . . . . . . . . . . . 12 ⊢ (𝜑 → (ω ∖ {∅}) ∈ 𝑈) |
7 | 4, 6 | eqeltrid 2842 | . . . . . . . . . . 11 ⊢ (𝜑 → N ∈ 𝑈) |
8 | 2, 7, 7 | wunxp 10667 | . . . . . . . . . 10 ⊢ (𝜑 → (N × N) ∈ 𝑈) |
9 | elpqn 10868 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
10 | 9 | ssriv 3953 | . . . . . . . . . . 11 ⊢ Q ⊆ (N × N) |
11 | 10 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → Q ⊆ (N × N)) |
12 | 2, 8, 11 | wunss 10655 | . . . . . . . . 9 ⊢ (𝜑 → Q ∈ 𝑈) |
13 | 2, 12 | wunpw 10650 | . . . . . . . 8 ⊢ (𝜑 → 𝒫 Q ∈ 𝑈) |
14 | prpssnq 10933 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → 𝑥 ⊊ Q) | |
15 | 14 | pssssd 4062 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ P → 𝑥 ⊆ Q) |
16 | velpw 4570 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 Q ↔ 𝑥 ⊆ Q) | |
17 | 15, 16 | sylibr 233 | . . . . . . . . . 10 ⊢ (𝑥 ∈ P → 𝑥 ∈ 𝒫 Q) |
18 | 17 | ssriv 3953 | . . . . . . . . 9 ⊢ P ⊆ 𝒫 Q |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → P ⊆ 𝒫 Q) |
20 | 2, 13, 19 | wunss 10655 | . . . . . . 7 ⊢ (𝜑 → P ∈ 𝑈) |
21 | 2, 20, 20 | wunxp 10667 | . . . . . 6 ⊢ (𝜑 → (P × P) ∈ 𝑈) |
22 | 2, 21 | wunpw 10650 | . . . . 5 ⊢ (𝜑 → 𝒫 (P × P) ∈ 𝑈) |
23 | enrer 11006 | . . . . . . 7 ⊢ ~R Er (P × P) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ~R Er (P × P)) |
25 | 24 | qsss 8724 | . . . . 5 ⊢ (𝜑 → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
26 | 2, 22, 25 | wunss 10655 | . . . 4 ⊢ (𝜑 → ((P × P) / ~R ) ∈ 𝑈) |
27 | 3, 26 | eqeltrid 2842 | . . 3 ⊢ (𝜑 → R ∈ 𝑈) |
28 | 2, 27, 27 | wunxp 10667 | . 2 ⊢ (𝜑 → (R × R) ∈ 𝑈) |
29 | 1, 28 | eqeltrid 2842 | 1 ⊢ (𝜑 → ℂ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∖ cdif 3912 ⊆ wss 3915 ∅c0 4287 𝒫 cpw 4565 {csn 4591 × cxp 5636 ωcom 7807 Er wer 8652 / cqs 8654 WUnicwun 10643 Ncnpi 10787 Qcnq 10795 Pcnp 10802 ~R cer 10807 Rcnr 10808 ℂcc 11056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-omul 8422 df-er 8655 df-ec 8657 df-qs 8661 df-wun 10645 df-ni 10815 df-pli 10816 df-mi 10817 df-lti 10818 df-plpq 10851 df-mpq 10852 df-ltpq 10853 df-enq 10854 df-nq 10855 df-erq 10856 df-plq 10857 df-mq 10858 df-1nq 10859 df-rq 10860 df-ltnq 10861 df-np 10924 df-plp 10926 df-ltp 10928 df-enr 10998 df-nr 10999 df-c 11064 |
This theorem is referenced by: wunndx 17074 |
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