Theorem lenlts 27894

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

Assertion

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

Proof of Theorem lenlts

Step	Hyp	Ref	Expression
1		df-les 27887	. . . 4 ⊢ ≤s = (( No × No ) ∖ ◡ <s )
2	1	breqi 5116	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ◡ <s )𝐵)
3		brdif 5165	. . 3 ⊢ (𝐴(( No × No ) ∖ ◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴◡ <s 𝐵))
4		brxp 5712	. . . 4 ⊢ (𝐴( No × No )𝐵 ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No ))
5	4	anbi1i 635	. . 3 ⊢ ((𝐴( No × No )𝐵 ∧ ¬ 𝐴◡ <s 𝐵) ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴◡ <s 𝐵))
6	2, 3, 5	3bitri 300	. 2 ⊢ (𝐴 ≤s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴◡ <s 𝐵))
7		ibar 537	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴◡ <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴◡ <s 𝐵)))
8		brcnvg 5867	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴◡ <s 𝐵 ↔ 𝐵 <s 𝐴))
9	8	notbid 321	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴◡ <s 𝐵 ↔ ¬ 𝐵 <s 𝐴))
10	7, 9	bitr3d 284	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴◡ <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
11	6, 10	bitrid 286	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   ∖ cdif 3903   class class class wbr 5110   × cxp 5661  ^◡ccnv 5662   No csur 27782   <s clts 27783   ≤s cles 27886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-les 27887

This theorem is referenced by:  ltnles  27895  lesloe  27896  lestri3  27897  lesnltd  27898  ltlestr  27902  leltstr  27903  lestr  27904  lesid  27909  lestric  27910  ltlesd  27915  ltsrec  27972  ltslpss  28079  cofcutr  28095  lenegs  28217  lesubsubsbd  28257  lesubsubs2bd  28258  lesubsubs3bd  28259  lesubaddsd  28264  lemuls2d  28345  lemuls1d  28346  ltonold  28432  oncutlt  28435  onnolt  28437  onles  28439  om2noseqlt2  28471  n0fincut  28526  bdaypw2n0bndlem  28634  bdaypw2bnd  28636  bdayfinbndlem1  28638  z12bdaylem1  28641