xgepnf - Metamath Proof Explorer

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Theorem xgepnf 13082

Description: An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)

**Assertion**
Ref	Expression
xgepnf	⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))

**Proof of Theorem xgepnf**
Step	Hyp	Ref	Expression
1		pnfxr 11188	. . 3 ⊢ +∞ ∈ ℝ^*
2		xrlenlt 11199	. . 3 ⊢ ((+∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
3	1, 2	mpan 691	. 2 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
4		nltpnft 13081	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
5	3, 4	bitr4d 282	1 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1542 ∈ wcel 2114 class class class wbr 5097 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-pre-lttri 11102 ax-pre-lttrn 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5518 df-po 5531 df-so 5532 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174

This theorem is referenced by: xnn0lenn0nn0 13162 xdivpnfrp 32993 xrge0npcan 33081 esumpinfval 34209 esumpinfsum 34213 esumpcvgval 34214 voliune 34365 volfiniune 34366