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Theorem xrltne 12897

Description: 'Less than' implies not equal for extended reals. (Contributed by NM, 20-Jan-2006.)

**Assertion**
Ref	Expression
xrltne	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴)

**Proof of Theorem xrltne**
Step	Hyp	Ref	Expression
1		orc 864	. . . 4 ⊢ (𝐴 < 𝐵 → (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))
2		xrltso 12875	. . . . . 6 ⊢ < Or ℝ^*
3		sotrieq 5532	. . . . . 6 ⊢ (( < Or ℝ^* ∧ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*)) → (𝐴 = 𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
4	2, 3	mpan 687	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 = 𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴)))
5	4	necon2abid 2986	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) ↔ 𝐴 ≠ 𝐵))
6	1, 5	syl5ib 243	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 ≠ 𝐵))
7	6	3impia 1116	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ≠ 𝐵)
8	7	necomd 2999	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 Or wor 5502 ℝ^*cxr 11008 < clt 11009

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014

This theorem is referenced by: xmulpnf1 13008 supxrbnd 13062 sgnp 14801 sgnn 14805 xrsdsreclblem 20644 supxrnemnf 31091 lfuhgr2 33080 acycgr2v 33112 xrgtned 42861 icoiccdif 43062 cncfiooicclem1 43434

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