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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 12957), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5281 in later theorems by invoking Axiom ax-cnex 11153 instead of cnexALT 12957. Use cnex 11178 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcnex | ⊢ ℂ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 11103 | . 2 ⊢ ℂ = (R × R) | |
2 | nrex1 11046 | . . 3 ⊢ R ∈ V | |
3 | 2, 2 | xpex 7727 | . 2 ⊢ (R × R) ∈ V |
4 | 1, 3 | eqeltri 2830 | 1 ⊢ ℂ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 × cxp 5670 Rcnr 10847 ℂcc 11095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-oadd 8457 df-omul 8458 df-er 8691 df-ec 8693 df-qs 8697 df-ni 10854 df-pli 10855 df-mi 10856 df-lti 10857 df-plpq 10890 df-mpq 10891 df-ltpq 10892 df-enq 10893 df-nq 10894 df-erq 10895 df-plq 10896 df-mq 10897 df-1nq 10898 df-rq 10899 df-ltnq 10900 df-np 10963 df-plp 10965 df-ltp 10967 df-enr 11037 df-nr 11038 df-c 11103 |
This theorem is referenced by: (None) |
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