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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33639 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴))) | |
2 | slenlt 33641 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) | |
3 | slenlt 33641 | . . . . 5 ⊢ ((𝐵 ∈ No ∧ 𝐴 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵)) | |
4 | 3 | ancoms 462 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐵 ≤s 𝐴 ↔ ¬ 𝐴 <s 𝐵)) |
5 | 2, 4 | anbi12d 634 | . . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴) ↔ (¬ 𝐵 <s 𝐴 ∧ ¬ 𝐴 <s 𝐵))) |
6 | ancom 464 | . . 3 ⊢ ((¬ 𝐵 <s 𝐴 ∧ ¬ 𝐴 <s 𝐵) ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴)) | |
7 | 5, 6 | bitrdi 290 | . 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴) ↔ (¬ 𝐴 <s 𝐵 ∧ ¬ 𝐵 <s 𝐴))) |
8 | 1, 7 | bitr4d 285 | 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 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 No csur 33529 <s cslt 33530 ≤s csle 33633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-1o 8180 df-2o 8181 df-no 33532 df-slt 33533 df-sle 33634 |
This theorem is referenced by: (None) |
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