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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 11166. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axicn | ⊢ i ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 11072 | . 2 ⊢ 0R ∈ R | |
2 | 1sr 11073 | . 2 ⊢ 1R ∈ R | |
3 | df-i 11116 | . . . 4 ⊢ i = ⟨0R, 1R⟩ | |
4 | 3 | eleq1i 2816 | . . 3 ⊢ (i ∈ ℂ ↔ ⟨0R, 1R⟩ ∈ ℂ) |
5 | opelcn 11121 | . . 3 ⊢ (⟨0R, 1R⟩ ∈ ℂ ↔ (0R ∈ R ∧ 1R ∈ R)) | |
6 | 4, 5 | bitri 275 | . 2 ⊢ (i ∈ ℂ ↔ (0R ∈ R ∧ 1R ∈ R)) |
7 | 1, 2, 6 | mpbir2an 708 | 1 ⊢ i ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2098 ⟨cop 4627 Rcnr 10857 0Rc0r 10858 1Rc1r 10859 ℂcc 11105 ici 11109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-omul 8467 df-er 8700 df-ec 8702 df-qs 8706 df-ni 10864 df-pli 10865 df-mi 10866 df-lti 10867 df-plpq 10900 df-mpq 10901 df-ltpq 10902 df-enq 10903 df-nq 10904 df-erq 10905 df-plq 10906 df-mq 10907 df-1nq 10908 df-rq 10909 df-ltnq 10910 df-np 10973 df-1p 10974 df-plp 10975 df-enr 11047 df-nr 11048 df-0r 11052 df-1r 11053 df-c 11113 df-i 11116 |
This theorem is referenced by: (None) |
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