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Mirrors > Home > MPE Home > Th. List > sletri3 | Structured version Visualization version GIF version |
Description: Trichotomy law for surreal less-than or equal. (Contributed by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
sletri3 | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slttrieq2 27810 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴))) | |
2 | slenlt 27812 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) | |
3 | slenlt 27812 | . . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵)) | |
4 | 3 | ancoms 458 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵)) |
5 | 2, 4 | anbi12d 632 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴) ↔ (¬ 𝐵 <s 𝐴 ∧ ¬ 𝐴 <s 𝐵))) |
6 | ancom 460 | . . 3 ⊢ ((¬ 𝐵 <s 𝐴 ∧ ¬ 𝐴 <s 𝐵) ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴)) | |
7 | 5, 6 | bitrdi 287 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴) ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴))) |
8 | 1, 7 | bitr4d 282 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 No csur 27699 <s cslt 27700 ≤s csle 27804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-sle 27805 |
This theorem is referenced by: addscan2 28041 mulscan2dlem 28219 n0subs 28375 nohalf 28422 |
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