Theorem addex 12969

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 11185	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 11187	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 7733	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 7917	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1457	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2098  Vcvv 3466   × cxp 5664  ⟶wf 6529  ℂcc 11104   + caddc 11109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-addf 11185

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-fun 6535  df-fn 6536  df-f 6537

This theorem is referenced by:  cnaddablx  19778  cnaddabl  19779  cnaddid  19780  cnaddinv  19781  zaddablx  19782  cnfldadd  21233  cnfldfun  21240  cnfldfunALT  21241  cnfldfunALTOLD  21242  cnlmodlem2  24986  cnnvg  30400  cnnvs  30402  cncph  30541  cnaddcom  38332  nn0mnd  47042