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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 12235), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5081 in later theorems by invoking the axiom ax-cnex 10439 instead of cnexALT 12235. Use cnex 10464 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcnex | ⊢ ℂ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 10389 | . 2 ⊢ ℂ = (R × R) | |
2 | nrex1 10332 | . . 3 ⊢ R ∈ V | |
3 | 2, 2 | xpex 7333 | . 2 ⊢ (R × R) ∈ V |
4 | 1, 3 | eqeltri 2879 | 1 ⊢ ℂ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2081 Vcvv 3437 × cxp 5441 Rcnr 10133 ℂcc 10381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-omul 7958 df-er 8139 df-ec 8141 df-qs 8145 df-ni 10140 df-pli 10141 df-mi 10142 df-lti 10143 df-plpq 10176 df-mpq 10177 df-ltpq 10178 df-enq 10179 df-nq 10180 df-erq 10181 df-plq 10182 df-mq 10183 df-1nq 10184 df-rq 10185 df-ltnq 10186 df-np 10249 df-plp 10251 df-ltp 10253 df-enr 10323 df-nr 10324 df-c 10389 |
This theorem is referenced by: (None) |
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