Theorem addex 13025

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 11237	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 11239	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 7761	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 7947	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1458	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2099  Vcvv 3462   × cxp 5680  ⟶wf 6550  ℂcc 11156   + caddc 11161

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 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-addf 11237

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-fun 6556  df-fn 6557  df-f 6558

This theorem is referenced by:  cnaddablx  19866  cnaddabl  19867  cnaddid  19868  cnaddinv  19869  zaddablx  19870  cnfldaddOLD  21363  cnfldfunOLD  21370  cnfldfunALTOLD  21371  cnfldfunALTOLDOLD  21372  cnlmodlem2  25155  cnnvg  30611  cnnvs  30613  cncph  30752  cnaddcom  38670  nn0mnd  47556