xgepnf - Metamath Proof Explorer

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Theorem xgepnf 13250

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 11320	. . 3 ⊢ +∞ ∈ ℝ^*
2		xrlenlt 11331	. . 3 ⊢ ((+∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
3	1, 2	mpan 703	. 2 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
4		nltpnft 13249	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
5	3, 4	bitr4d 285	1 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 +∞cpnf 11297 ℝ^*cxr 11299 < clt 11300 ≤ cle 11301

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-pre-lttri 11231 ax-pre-lttrn 11232

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5543 df-po 5556 df-so 5557 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306

This theorem is used by: xnn0lenn0nn0 13330 xdivpnfrp 33418 xrge0npcan 33500 esumpinfval 34624 esumpinfsum 34628 esumpcvgval 34629 voliune 34781 volfiniune 34782

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