Theorem addex 13054

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 11263	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 11265	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 7788	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 7974	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1461	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3488   × cxp 5698  ⟶wf 6569  ℂcc 11182   + caddc 11187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-addf 11263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  cnaddablx  19910  cnaddabl  19911  cnaddid  19912  cnaddinv  19913  zaddablx  19914  cnfldaddOLD  21407  cnfldfunOLD  21414  cnfldfunALTOLD  21415  cnfldfunALTOLDOLD  21416  cnlmodlem2  25189  cnnvg  30710  cnnvs  30712  cncph  30851  cnaddcom  38928  nn0mnd  47902