slenlt - Metamath Proof Explorer

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Theorem slenlt 27812

Description: Surreal less-than or equal in terms of less-than. (Contributed by Scott Fenton, 8-Dec-2021.)

**Assertion**
Ref	Expression
slenlt	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

**Proof of Theorem slenlt**
Step	Hyp	Ref	Expression
1		df-sle 27805	. . . 4 ⊢ ≤s = (( No × No ) ∖ ^◡ <s )
2	1	breqi 5154	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ^◡ <s )𝐵)
3		brdif 5201	. . 3 ⊢ (𝐴(( No × No ) ∖ ^◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴^◡ <s 𝐵))
4		brxp 5738	. . . 4 ⊢ (𝐴( No × No )𝐵 ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No ))
5	4	anbi1i 624	. . 3 ⊢ ((𝐴( No × No )𝐵 ∧ ¬ 𝐴^◡ <s 𝐵) ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵))
6	2, 3, 5	3bitri 297	. 2 ⊢ (𝐴 ≤s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵))
7		ibar 528	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴^◡ <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵)))
8		brcnvg 5893	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴^◡ <s 𝐵 ↔ 𝐵 <s 𝐴))
9	8	notbid 318	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴^◡ <s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
10	7, 9	bitr3d 281	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
11	6, 10	bitrid 283	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∖ cdif 3960 class class class wbr 5148 × cxp 5687 ^◡ccnv 5688 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-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-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-sle 27805

This theorem is referenced by: sltnle 27813 sleloe 27814 sletri3 27815 sltletr 27816 slelttr 27817 sletr 27818 slerflex 27823 sletric 27824 sltled 27829 sltrec 27880 sltlpss 27960 cofcutr 27973 sleneg 28093 slesubsubbd 28131 slesubsub2bd 28132 slesubsub3bd 28133 slesubaddd 28138 slemul2d 28215 slemul1d 28216 sltonold 28298 om2noseqlt2 28321

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