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Theorem slenlt 27182

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 27175	. . . 4 ⊢ ≤s = (( No × No ) ∖ ^◡ <s )
2	1	breqi 5147	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ^◡ <s )𝐵)
3		brdif 5194	. . 3 ⊢ (𝐴(( No × No ) ∖ ^◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴^◡ <s 𝐵))
4		brxp 5717	. . . 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 5871	. . . 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 3941 class class class wbr 5141 × cxp 5667 ^◡ccnv 5668 No csur 27070 <s cslt 27071 ≤s csle 27174

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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420

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 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-sle 27175

This theorem is referenced by: sltnle 27183 sleloe 27184 sletri3 27185 sltletr 27186 slelttr 27187 sletr 27188 slerflex 27193 sletric 27194 sltled 27199 sltrec 27247 sltlpss 27324 cofcutr 27331 sleneg 27436 slesubsubbd 27467 slesubsub2bd 27468 slesubsub3bd 27469 slemul2d 27537 slemul1d 27538

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