Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axcnex Structured version   Visualization version   GIF version

Theorem axcnex 9847
 Description: The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 11704), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4699 in later theorems by invoking the axiom ax-cnex 9871 instead of cnexALT 11704. Use cnex 9896 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
Assertion
Ref Expression
axcnex ℂ ∈ V

Proof of Theorem axcnex
StepHypRef Expression
1 df-c 9821 . 2 ℂ = (R × R)
2 df-nr 9757 . . . 4 R = ((P × P) / ~R )
3 npex 9687 . . . . . . 7 P ∈ V
43, 3xpex 6860 . . . . . 6 (P × P) ∈ V
54pwex 4774 . . . . 5 𝒫 (P × P) ∈ V
6 enrer 9765 . . . . . . . 8 ~R Er (P × P)
76a1i 11 . . . . . . 7 (⊤ → ~R Er (P × P))
87qsss 7695 . . . . . 6 (⊤ → ((P × P) / ~R ) ⊆ 𝒫 (P × P))
98trud 1484 . . . . 5 ((P × P) / ~R ) ⊆ 𝒫 (P × P)
105, 9ssexi 4731 . . . 4 ((P × P) / ~R ) ∈ V
112, 10eqeltri 2684 . . 3 R ∈ V
1211, 11xpex 6860 . 2 (R × R) ∈ V
131, 12eqeltri 2684 1 ℂ ∈ V
 Colors of variables: wff setvar class Syntax hints:  ⊤wtru 1476   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108   × cxp 5036   Er wer 7626   / cqs 7628  Pcnp 9560   ~R cer 9565  Rcnr 9566  ℂcc 9813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-ltp 9686  df-enr 9756  df-nr 9757  df-c 9821 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator