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Mirrors > Home > ILE Home > Th. List > axcnex | GIF version |
Description: The complex numbers form a set. Use cnex 7856 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcnex | ⊢ ℂ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 7738 | . 2 ⊢ ℂ = (R × R) | |
2 | df-nr 7647 | . . . 4 ⊢ R = ((P × P) / ~R ) | |
3 | npex 7393 | . . . . . . 7 ⊢ P ∈ V | |
4 | 3, 3 | xpex 4701 | . . . . . 6 ⊢ (P × P) ∈ V |
5 | 4 | pwex 4144 | . . . . 5 ⊢ 𝒫 (P × P) ∈ V |
6 | enrer 7655 | . . . . . . . 8 ⊢ ~R Er (P × P) | |
7 | 6 | a1i 9 | . . . . . . 7 ⊢ (⊤ → ~R Er (P × P)) |
8 | 7 | qsss 6539 | . . . . . 6 ⊢ (⊤ → ((P × P) / ~R ) ⊆ 𝒫 (P × P)) |
9 | 8 | mptru 1344 | . . . . 5 ⊢ ((P × P) / ~R ) ⊆ 𝒫 (P × P) |
10 | 5, 9 | ssexi 4102 | . . . 4 ⊢ ((P × P) / ~R ) ∈ V |
11 | 2, 10 | eqeltri 2230 | . . 3 ⊢ R ∈ V |
12 | 11, 11 | xpex 4701 | . 2 ⊢ (R × R) ∈ V |
13 | 1, 12 | eqeltri 2230 | 1 ⊢ ℂ ∈ V |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1336 ∈ wcel 2128 Vcvv 2712 ⊆ wss 3102 𝒫 cpw 3543 × cxp 4584 Er wer 6477 / cqs 6479 Pcnp 7211 ~R cer 7216 Rcnr 7217 ℂcc 7730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-2o 6364 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 df-ltnqqs 7273 df-enq0 7344 df-nq0 7345 df-0nq0 7346 df-plq0 7347 df-mq0 7348 df-inp 7386 df-iplp 7388 df-enr 7646 df-nr 7647 df-c 7738 |
This theorem is referenced by: peano5nnnn 7812 |
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