xgepnf - Metamath Proof Explorer

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Theorem xgepnf 13170

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 11238	. . 3 ⊢ +∞ ∈ ℝ^*
2		xrlenlt 11249	. . 3 ⊢ ((+∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
3	1, 2	mpan 700	. 2 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
4		nltpnft 13169	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
5	3, 4	bitr4d 284	1 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1562 ∈ wcel 2144 class class class wbr 5102 +∞cpnf 11215 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-pre-lttri 11149 ax-pre-lttrn 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-po 5557 df-so 5558 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224

This theorem is referenced by: xnn0lenn0nn0 13250 xdivpnfrp 33112 xrge0npcan 33200 esumpinfval 34372 esumpinfsum 34376 esumpcvgval 34377 voliune 34528 volfiniune 34529