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Mirrors > Home > MPE Home > Th. List > mins2 | Structured version Visualization version GIF version |
Description: The minimum of two surreals is less than or equal to the second. (Contributed by Scott Fenton, 14-Feb-2025.) |
Ref | Expression |
---|---|
mins2 | ⊢ (𝐵 ∈ No → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slerflex 27636 | . . 3 ⊢ (𝐵 ∈ No → 𝐵 ≤s 𝐵) | |
2 | iffalse 4530 | . . . 4 ⊢ (¬ 𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐵) | |
3 | 2 | breq1d 5149 | . . 3 ⊢ (¬ 𝐴 ≤s 𝐵 → (if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵 ↔ 𝐵 ≤s 𝐵)) |
4 | 1, 3 | syl5ibrcom 246 | . 2 ⊢ (𝐵 ∈ No → (¬ 𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵)) |
5 | iftrue 4527 | . . 3 ⊢ (𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) = 𝐴) | |
6 | id 22 | . . 3 ⊢ (𝐴 ≤s 𝐵 → 𝐴 ≤s 𝐵) | |
7 | 5, 6 | eqbrtrd 5161 | . 2 ⊢ (𝐴 ≤s 𝐵 → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵) |
8 | 4, 7 | pm2.61d2 181 | 1 ⊢ (𝐵 ∈ No → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2098 ifcif 4521 class class class wbr 5139 No csur 27513 ≤s csle 27617 |
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 5290 ax-nul 5297 ax-pr 5418 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-1o 8462 df-2o 8463 df-no 27516 df-slt 27517 df-sle 27618 |
This theorem is referenced by: (None) |
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