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Mirrors > Home > MPE Home > Th. List > xrltnsym2 | Structured version Visualization version GIF version |
Description: 'Less than' is antisymmetric and irreflexive for extended reals. (Contributed by NM, 6-Feb-2007.) |
Ref | Expression |
---|---|
xrltnsym2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltnsym 12369 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 → ¬ 𝐵 < 𝐴)) | |
2 | imnan 400 | . 2 ⊢ ((𝐴 < 𝐵 → ¬ 𝐵 < 𝐴) ↔ ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴)) | |
3 | 1, 2 | sylib 219 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ (𝐴 < 𝐵 ∧ 𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2079 class class class wbr 4956 ℝ*cxr 10509 < clt 10510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-pre-lttri 10446 ax-pre-lttrn 10447 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 |
This theorem is referenced by: (None) |
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