xgepnf - Metamath Proof Explorer

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Theorem xgepnf 13094

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 11200	. . 3 ⊢ +∞ ∈ ℝ^*
2		xrlenlt 11211	. . 3 ⊢ ((+∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
3	1, 2	mpan 691	. 2 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
4		nltpnft 13093	. 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 5100 +∞cpnf 11177 ℝ^*cxr 11179 < clt 11180 ≤ cle 11181

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 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-pre-lttri 11114 ax-pre-lttrn 11115

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-po 5542 df-so 5543 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186

This theorem is referenced by: xnn0lenn0nn0 13174 xdivpnfrp 33031 xrge0npcan 33119 esumpinfval 34257 esumpinfsum 34261 esumpcvgval 34262 voliune 34413 volfiniune 34414