Theorem pnfex 11268

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 11251	. 2 ⊢ +∞ = 𝒫 ∪ ℂ
2		cnex 11187	. . . 4 ⊢ ℂ ∈ V
3	2	uniex 7741	. . 3 ⊢ ∪ ℂ ∈ V
4	3	pwex 5350	. 2 ⊢ 𝒫 ∪ ℂ ∈ V
5	1, 4	eqeltri 2858	1 ⊢ +∞ ∈ V

This proof depends on syntax axioms:   ∈ wcel 2142  Vcvv 3454  𝒫 cpw 4561  ∪ cuni 4871  ℂcc 11104  +∞cpnf 11246

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pow 5335  ax-un 7734  ax-cnex 11162

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-ss 3921  df-pw 4563  df-uni 4872  df-pnf 11251

This theorem is used by:  pnfxr  11269  mnfxr  11272  elxnn0  12585  elxr  13147  xnegex  13240  xaddval  13255  xmulval  13257  xrinfmss  13342  hashgval  14376  hashinf  14378  hashfxnn0  14380  pcval  16910  pc0  16920  ramcl2  17082  iccpnfhmeo  25115  taylfval  26533  xrlimcnp  27144  xrge0iifcv  34333  xrge0iifiso  34334  xrge0iifhom  34336  sge0f1o  47124  sge0sup  47133  sge0pnfmpt  47187