Theorem addex 12930

Description: The addition operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
addex	⊢ + ∈ V

Proof of Theorem addex

Step	Hyp	Ref	Expression
1		ax-addf 11108	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 11110	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 7700	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 7880	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1464	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3430   × cxp 5622  ⟶wf 6488  ℂcc 11027   + caddc 11032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-addf 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  cnaddablx  19834  cnaddabl  19835  cnaddid  19836  cnaddinv  19837  zaddablx  19838  cnfldaddOLD  21364  cnfldfunOLD  21371  cnfldfunALTOLD  21372  cnlmodlem2  25114  cnnvg  30764  cnnvs  30766  cncph  30905  cnaddcom  39432  nn0mnd  48667