slenlt - Metamath Proof Explorer

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Theorem slenlt 27255

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 27248	. . . 4 ⊢ ≤s = (( No × No ) ∖ ^◡ <s )
2	1	breqi 5155	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ^◡ <s )𝐵)
3		brdif 5202	. . 3 ⊢ (𝐴(( No × No ) ∖ ^◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴^◡ <s 𝐵))
4		brxp 5726	. . . 4 ⊢ (𝐴( No × No )𝐵 ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No ))
5	4	anbi1i 625	. . 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 530	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴^◡ <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴^◡ <s 𝐵)))
8		brcnvg 5880	. . . 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 205 ∧ wa 397 ∈ wcel 2107 ∖ cdif 3946 class class class wbr 5149 × cxp 5675 ^◡ccnv 5676 No csur 27143 <s cslt 27144 ≤s csle 27247

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-cnv 5685 df-sle 27248

This theorem is referenced by: sltnle 27256 sleloe 27257 sletri3 27258 sltletr 27259 slelttr 27260 sletr 27261 slerflex 27266 sletric 27267 sltled 27272 sltrec 27321 sltlpss 27401 cofcutr 27411 sleneg 27520 slesubsubbd 27553 slesubsub2bd 27554 slesubsub3bd 27555 slemul2d 27626 slemul1d 27627

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