|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > pnfex | Structured version Visualization version GIF version | ||
| Description: Plus infinity exists. (Contributed by David A. Wheeler, 8-Dec-2018.) (Revised by Steven Nguyen, 7-Dec-2022.) | 
| Ref | Expression | 
|---|---|
| pnfex | ⊢ +∞ ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-pnf 11297 | . 2 ⊢ +∞ = 𝒫 ∪ ℂ | |
| 2 | cnex 11236 | . . . 4 ⊢ ℂ ∈ V | |
| 3 | 2 | uniex 7761 | . . 3 ⊢ ∪ ℂ ∈ V | 
| 4 | 3 | pwex 5380 | . 2 ⊢ 𝒫 ∪ ℂ ∈ V | 
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ +∞ ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 Vcvv 3480 𝒫 cpw 4600 ∪ cuni 4907 ℂcc 11153 +∞cpnf 11292 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-pow 5365 ax-un 7755 ax-cnex 11211 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-pw 4602 df-uni 4908 df-pnf 11297 | 
| This theorem is referenced by: pnfxr 11315 mnfxr 11318 elxnn0 12601 elxr 13158 xnegex 13250 xaddval 13265 xmulval 13267 xrinfmss 13352 hashgval 14372 hashinf 14374 hashfxnn0 14376 pcval 16882 pc0 16892 ramcl2 17054 iccpnfhmeo 24976 taylfval 26400 xrlimcnp 27011 xrge0iifcv 33933 xrge0iifiso 33934 xrge0iifhom 33936 sge0f1o 46397 sge0sup 46406 sge0pnfmpt 46460 | 
| Copyright terms: Public domain | W3C validator |