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Mirrors > Home > MPE Home > Th. List > mins1 | Structured version Visualization version GIF version |
Description: The minimum of two surreals is less than or equal to the first. (Contributed by Scott Fenton, 14-Feb-2025.) |
Ref | Expression |
---|---|
mins1 | ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4526 | . . . 4 ⊢ (𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐴) | |
2 | 1 | adantl 481 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐴) |
3 | slerflex 27600 | . . . 4 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴) | |
4 | 3 | ad2antrr 723 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 𝐵) → 𝐴 ≤s 𝐴) |
5 | 2, 4 | eqbrtrd 5160 | . 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴) |
6 | iffalse 4529 | . . . 4 ⊢ (¬ 𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐵) | |
7 | 6 | adantl 481 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐵) |
8 | sletric 27601 | . . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴)) | |
9 | 8 | orcanai 999 | . . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴 ≤s 𝐵) → 𝐵 ≤s 𝐴) |
10 | 7, 9 | eqbrtrd 5160 | . 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴 ≤s 𝐵) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴) |
11 | 5, 10 | pm2.61dan 810 | 1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ifcif 4520 class class class wbr 5138 No csur 27477 ≤s csle 27581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 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-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-1o 8461 df-2o 8462 df-no 27480 df-slt 27481 df-sle 27582 |
This theorem is referenced by: (None) |
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