Theorem pnfex 11317

Description: Plus infinity exists. (Contributed by David A. Wheeler, 8-Dec-2018.) (Revised by Steven Nguyen, 7-Dec-2022.)

Assertion

Ref	Expression
pnfex	⊢ +∞ ∈ V

Proof of Theorem pnfex

Step	Hyp	Ref	Expression
1		df-pnf 11300	. 2 ⊢ +∞ = 𝒫 ∪ ℂ
2		cnex 11239	. . . 4 ⊢ ℂ ∈ V
3	2	uniex 7752	. . 3 ⊢ ∪ ℂ ∈ V
4	3	pwex 5384	. 2 ⊢ 𝒫 ∪ ℂ ∈ V
5	1, 4	eqeltri 2822	1 ⊢ +∞ ∈ V

Syntax hints:   ∈ wcel 2099  Vcvv 3462  𝒫 cpw 4607  ∪ cuni 4913  ℂcc 11156  +∞cpnf 11295

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-pow 5369  ax-un 7746  ax-cnex 11214

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-ss 3964  df-pw 4609  df-uni 4914  df-pnf 11300

This theorem is referenced by:  pnfxr  11318  mnfxr  11321  elxnn0  12598  elxr  13150  xnegex  13241  xaddval  13256  xmulval  13258  xrinfmss  13343  hashgval  14350  hashinf  14352  hashfxnn0  14354  pcval  16846  pc0  16856  ramcl2  17018  iccpnfhmeo  24961  taylfval  26386  xrlimcnp  26996  xrge0iifcv  33749  xrge0iifiso  33750  xrge0iifhom  33752  sge0f1o  46003  sge0sup  46012  sge0pnfmpt  46066