xrletri - Metamath Proof Explorer

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Theorem xrletri 13196

Description: Trichotomy law for extended reals. (Contributed by NM, 7-Feb-2007.)

**Assertion**
Ref	Expression
xrletri	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))

**Proof of Theorem xrletri**
Step	Hyp	Ref	Expression
1		xrltnle 11294	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵))
2	1	ancoms 464	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐵))
3		xrltle 13192	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 < 𝐴 → 𝐵 ≤ 𝐴))
4	3	ancoms 464	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 < 𝐴 → 𝐵 ≤ 𝐴))
5	2, 4	sylbird 263	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 𝐴 ≤ 𝐵 → 𝐵 ≤ 𝐴))
6	5	orrd 877	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∈ wcel 2146 class class class wbr 5114 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-pre-lttri 11192 ax-pre-lttrn 11193

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-po 5574 df-so 5575 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267

This theorem is used by: xrmax2 13220 xrmin1 13221 pcgcd 16963 letsr 18674 xrsdsreclb 21601 xrge0omnd 21632 xrsxmet 25004 xrhmeo 25142 itg2const2 25937 xrge0infss 33142 xrstos 33361

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