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Theorem slenlt 27479

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 27472	. . . 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 5725	. . . 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 296	. 2 ⊢ (𝐴 ≤s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵))
7		ibar 529	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴^◡ <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵)))
8		brcnvg 5879	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴^◡ <s 𝐵 ↔ 𝐵 <s 𝐴))
9	8	notbid 317	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴^◡ <s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
10	7, 9	bitr3d 280	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
11	6, 10	bitrid 282	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∖ cdif 3945 class class class wbr 5148 × cxp 5674 ^◡ccnv 5675 No csur 27367 <s cslt 27368 ≤s csle 27471

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-sle 27472

This theorem is referenced by: sltnle 27480 sleloe 27481 sletri3 27482 sltletr 27483 slelttr 27484 sletr 27485 slerflex 27490 sletric 27491 sltled 27496 sltrec 27546 sltlpss 27626 cofcutr 27637 sleneg 27747 slesubsubbd 27780 slesubsub2bd 27781 slesubsub3bd 27782 slemul2d 27853 slemul1d 27854 sltonold 27914

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