Theorem addex 13029

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 11232	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 11234	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 7772	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 7957	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1460	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3478   × cxp 5687  ⟶wf 6559  ℂcc 11151   + caddc 11156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567

This theorem is referenced by:  cnaddablx  19901  cnaddabl  19902  cnaddid  19903  cnaddinv  19904  zaddablx  19905  cnfldaddOLD  21402  cnfldfunOLD  21409  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411  cnlmodlem2  25184  cnnvg  30707  cnnvs  30709  cncph  30848  cnaddcom  38954  nn0mnd  48023