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Mirrors > Home > MPE Home > Th. List > axcnex | Structured version Visualization version GIF version |
Description: The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 12388), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5193 in later theorems by invoking the axiom ax-cnex 10596 instead of cnexALT 12388. Use cnex 10621 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcnex | ⊢ ℂ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 10546 | . 2 ⊢ ℂ = (R × R) | |
2 | nrex1 10489 | . . 3 ⊢ R ∈ V | |
3 | 2, 2 | xpex 7479 | . 2 ⊢ (R × R) ∈ V |
4 | 1, 3 | eqeltri 2912 | 1 ⊢ ℂ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3497 × cxp 5556 Rcnr 10290 ℂcc 10538 |
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 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-ec 8294 df-qs 8298 df-ni 10297 df-pli 10298 df-mi 10299 df-lti 10300 df-plpq 10333 df-mpq 10334 df-ltpq 10335 df-enq 10336 df-nq 10337 df-erq 10338 df-plq 10339 df-mq 10340 df-1nq 10341 df-rq 10342 df-ltnq 10343 df-np 10406 df-plp 10408 df-ltp 10410 df-enr 10480 df-nr 10481 df-c 10546 |
This theorem is referenced by: (None) |
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