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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 10589. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axicn | ⊢ i ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0r 10495 | . 2 ⊢ 0R ∈ R | |
2 | 1sr 10496 | . 2 ⊢ 1R ∈ R | |
3 | df-i 10539 | . . . 4 ⊢ i = 〈0R, 1R〉 | |
4 | 3 | eleq1i 2902 | . . 3 ⊢ (i ∈ ℂ ↔ 〈0R, 1R〉 ∈ ℂ) |
5 | opelcn 10544 | . . 3 ⊢ (〈0R, 1R〉 ∈ ℂ ↔ (0R ∈ R ∧ 1R ∈ R)) | |
6 | 4, 5 | bitri 277 | . 2 ⊢ (i ∈ ℂ ↔ (0R ∈ R ∧ 1R ∈ R)) |
7 | 1, 2, 6 | mpbir2an 709 | 1 ⊢ i ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2113 〈cop 4566 Rcnr 10280 0Rc0r 10281 1Rc1r 10282 ℂcc 10528 ici 10532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-omul 8100 df-er 8282 df-ec 8284 df-qs 8288 df-ni 10287 df-pli 10288 df-mi 10289 df-lti 10290 df-plpq 10323 df-mpq 10324 df-ltpq 10325 df-enq 10326 df-nq 10327 df-erq 10328 df-plq 10329 df-mq 10330 df-1nq 10331 df-rq 10332 df-ltnq 10333 df-np 10396 df-1p 10397 df-plp 10398 df-enr 10470 df-nr 10471 df-0r 10475 df-1r 10476 df-c 10536 df-i 10539 |
This theorem is referenced by: (None) |
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