slenlt - Metamath Proof Explorer

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Theorem slenlt 27662

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 27655	. . . 4 ⊢ ≤s = (( No × No ) ∖ ^◡ <s )
2	1	breqi 5098	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ^◡ <s )𝐵)
3		brdif 5145	. . 3 ⊢ (𝐴(( No × No ) ∖ ^◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴^◡ <s 𝐵))
4		brxp 5668	. . . 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 5822	. . . 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 2109 ∖ cdif 3900 class class class wbr 5092 × cxp 5617 ^◡ccnv 5618 No csur 27549 <s cslt 27550 ≤s csle 27654

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-br 5093 df-opab 5155 df-xp 5625 df-cnv 5627 df-sle 27655

This theorem is referenced by: sltnle 27663 sleloe 27664 sletri3 27665 sltletr 27666 slelttr 27667 sletr 27668 slerflex 27673 sletric 27674 sltled 27679 sltrec 27733 sltlpss 27824 cofcutr 27839 sleneg 27959 slesubsubbd 27997 slesubsub2bd 27998 slesubsub3bd 27999 slesubaddd 28004 slemul2d 28084 slemul1d 28085 sltonold 28169 onscutlt 28172 onnolt 28174 om2noseqlt2 28201 n0sfincut 28253

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