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Mirrors > Home > MPE Home > Th. List > axicn | Structured version Visualization version GIF version |
Description: i is a complex number. Axiom 3 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 10318. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axicn | ⊢ i ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 10224 | . 2 ⊢ 0R ∈ R | |
2 | 1sr 10225 | . 2 ⊢ 1R ∈ R | |
3 | df-i 10268 | . . . 4 ⊢ i = 〈0R, 1R〉 | |
4 | 3 | eleq1i 2897 | . . 3 ⊢ (i ∈ ℂ ↔ 〈0R, 1R〉 ∈ ℂ) |
5 | opelcn 10273 | . . 3 ⊢ (〈0R, 1R〉 ∈ ℂ ↔ (0R ∈ R ∧ 1R ∈ R)) | |
6 | 4, 5 | bitri 267 | . 2 ⊢ (i ∈ ℂ ↔ (0R ∈ R ∧ 1R ∈ R)) |
7 | 1, 2, 6 | mpbir2an 702 | 1 ⊢ i ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∈ wcel 2164 〈cop 4405 Rcnr 10009 0Rc0r 10010 1Rc1r 10011 ℂcc 10257 ici 10261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-omul 7836 df-er 8014 df-ec 8016 df-qs 8020 df-ni 10016 df-pli 10017 df-mi 10018 df-lti 10019 df-plpq 10052 df-mpq 10053 df-ltpq 10054 df-enq 10055 df-nq 10056 df-erq 10057 df-plq 10058 df-mq 10059 df-1nq 10060 df-rq 10061 df-ltnq 10062 df-np 10125 df-1p 10126 df-plp 10127 df-enr 10199 df-nr 10200 df-0r 10204 df-1r 10205 df-c 10265 df-i 10268 |
This theorem is referenced by: (None) |
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