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Mirrors > Home > ILE Home > Th. List > xrletrid | GIF version |
Description: Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrletrid.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xrletrid.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
xrletrid.3 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
xrletrid.4 | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
Ref | Expression |
---|---|
xrletrid | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrletrid.3 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | xrletrid.4 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
3 | xrletrid.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | xrletrid.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
5 | xrletri3 9840 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) | |
6 | 3, 4, 5 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴))) |
7 | 1, 2, 6 | mpbir2and 946 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 ℝ*cxr 8026 ≤ cle 8028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-pre-ltirr 7958 ax-pre-apti 7961 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-xp 4653 df-cnv 4655 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 |
This theorem is referenced by: pcadd2 12384 |
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