Theorem addex 12914

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 11117	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 11119	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 7708	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 7888	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1464	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3442   × cxp 5630  ⟶wf 6496  ℂcc 11036   + caddc 11041

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 5243  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-addf 11117

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by:  cnaddablx  19809  cnaddabl  19810  cnaddid  19811  cnaddinv  19812  zaddablx  19813  cnfldaddOLD  21341  cnfldfunOLD  21348  cnfldfunALTOLD  21349  cnlmodlem2  25105  cnnvg  30765  cnnvs  30767  cncph  30906  cnaddcom  39342  nn0mnd  48533