Theorem addex 9891

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 8159	. 2 ⊢ + :(ℂ × ℂ)⟶ℂ
2		cnex 8161	. . 3 ⊢ ℂ ∈ V
3	2, 2	xpex 4844	. 2 ⊢ (ℂ × ℂ) ∈ V
4		fex2 5505	. 2 ⊢ (( + :(ℂ × ℂ)⟶ℂ ∧ (ℂ × ℂ) ∈ V ∧ ℂ ∈ V) → + ∈ V)
5	1, 3, 2, 4	mp3an 1373	1 ⊢ + ∈ V

Syntax hints:   ∈ wcel 2201  Vcvv 2801   × cxp 4725  ⟶wf 5324  ℂcc 8035   + caddc 8040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-addf 8159

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-xp 4733  df-rel 4734  df-cnv 4735  df-dm 4737  df-rn 4738  df-fun 5330  df-fn 5331  df-f 5332

This theorem is referenced by: (None)