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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 13001), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 5285 in later theorems by invoking Axiom ax-cnex 11195 instead of cnexALT 13001. Use cnex 11220 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axcnex | ⊢ ℂ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-c 11145 | . 2 ⊢ ℂ = (R × R) | |
2 | nrex1 11088 | . . 3 ⊢ R ∈ V | |
3 | 2, 2 | xpex 7755 | . 2 ⊢ (R × R) ∈ V |
4 | 1, 3 | eqeltri 2825 | 1 ⊢ ℂ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3471 × cxp 5676 Rcnr 10889 ℂcc 11137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-ni 10896 df-pli 10897 df-mi 10898 df-lti 10899 df-plpq 10932 df-mpq 10933 df-ltpq 10934 df-enq 10935 df-nq 10936 df-erq 10937 df-plq 10938 df-mq 10939 df-1nq 10940 df-rq 10941 df-ltnq 10942 df-np 11005 df-plp 11007 df-ltp 11009 df-enr 11079 df-nr 11080 df-c 11145 |
This theorem is referenced by: (None) |
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