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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 10545 | . 2 ⊢ ℂ = (R × R) | |
2 | wuncn.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | df-nr 10480 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
4 | df-ni 10296 | . . . . . . . . . . . 12 ⊢ N = (ω ∖ {∅}) | |
5 | wuncn.2 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ω ∈ 𝑈) | |
6 | 2, 5 | wundif 10138 | . . . . . . . . . . . 12 ⊢ (𝜑 → (ω ∖ {∅}) ∈ 𝑈) |
7 | 4, 6 | eqeltrid 2919 | . . . . . . . . . . 11 ⊢ (𝜑 → N ∈ 𝑈) |
8 | 2, 7, 7 | wunxp 10148 | . . . . . . . . . 10 ⊢ (𝜑 → (N × N) ∈ 𝑈) |
9 | elpqn 10349 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → 𝑥 ∈ (N × N)) | |
10 | 9 | ssriv 3973 | . . . . . . . . . . 11 ⊢ Q ⊆ (N × N) |
11 | 10 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → Q ⊆ (N × N)) |
12 | 2, 8, 11 | wunss 10136 | . . . . . . . . 9 ⊢ (𝜑 → Q ∈ 𝑈) |
13 | 2, 12 | wunpw 10131 | . . . . . . . 8 ⊢ (𝜑 → 𝒫 Q ∈ 𝑈) |
14 | prpssnq 10414 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ P → 𝑥 ⊊ Q) | |
15 | 14 | pssssd 4076 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ P → 𝑥 ⊆ Q) |
16 | velpw 4546 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 Q ↔ 𝑥 ⊆ Q) | |
17 | 15, 16 | sylibr 236 | . . . . . . . . . 10 ⊢ (𝑥 ∈ P → 𝑥 ∈ 𝒫 Q) |
18 | 17 | ssriv 3973 | . . . . . . . . 9 ⊢ P ⊆ 𝒫 Q |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → P ⊆ 𝒫 Q) |
20 | 2, 13, 19 | wunss 10136 | . . . . . . 7 ⊢ (𝜑 → P ∈ 𝑈) |
21 | 2, 20, 20 | wunxp 10148 | . . . . . 6 ⊢ (𝜑 → (P × P) ∈ 𝑈) |
22 | 2, 21 | wunpw 10131 | . . . . 5 ⊢ (𝜑 → 𝒫 (P × P) ∈ 𝑈) |
23 | enrer 10487 | . . . . . . 7 ⊢ ~R Er (P × P) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ~R Er (P × P)) |
25 | 24 | qsss 8360 | . . . . 5 ⊢ (𝜑 → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
26 | 2, 22, 25 | wunss 10136 | . . . 4 ⊢ (𝜑 → ((P × P) / ~R ) ∈ 𝑈) |
27 | 3, 26 | eqeltrid 2919 | . . 3 ⊢ (𝜑 → R ∈ 𝑈) |
28 | 2, 27, 27 | wunxp 10148 | . 2 ⊢ (𝜑 → (R × R) ∈ 𝑈) |
29 | 1, 28 | eqeltrid 2919 | 1 ⊢ (𝜑 → ℂ ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {csn 4569 × cxp 5555 ωcom 7582 Er wer 8288 / cqs 8290 WUnicwun 10124 Ncnpi 10268 Qcnq 10276 Pcnp 10283 ~R cer 10288 Rcnr 10289 ℂcc 10537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-wun 10126 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 df-np 10405 df-plp 10407 df-ltp 10409 df-enr 10479 df-nr 10480 df-c 10545 |
This theorem is referenced by: wunndx 16506 |
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