Theorem addex 12995

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 11209	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 11211	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 7749	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 7935	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1458	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2099  Vcvv 3469   × cxp 5670  ⟶wf 6538  ℂcc 11128   + caddc 11133

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-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-addf 11209

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-fun 6544  df-fn 6545  df-f 6546

This theorem is referenced by:  cnaddablx  19814  cnaddabl  19815  cnaddid  19816  cnaddinv  19817  zaddablx  19818  cnfldaddOLD  21286  cnfldfunOLD  21293  cnfldfunALTOLD  21294  cnfldfunALTOLDOLD  21295  cnlmodlem2  25051  cnnvg  30475  cnnvs  30477  cncph  30616  cnaddcom  38381  nn0mnd  47164