Theorem lenlts 27874

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 27867	. . . 4 ⊢ ≤s = (( No × No ) ∖ ◡ <s )
2	1	breqi 5111	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ◡ <s )𝐵)
3		brdif 5158	. . 3 ⊢ (𝐴(( No × No ) ∖ ◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴◡ <s 𝐵))
4		brxp 5701	. . . 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 5856	. . . 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 2145   ∖ cdif 3904   class class class wbr 5105   × cxp 5650  ^◡ccnv 5651   No csur 27762   <s clts 27763   ≤s cles 27866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-les 27867

This theorem is referenced by:  ltnles  27875  lesloe  27876  lestri3  27877  lesnltd  27878  ltlestr  27882  leltstr  27883  lestr  27884  lesid  27889  lestric  27890  ltlesd  27895  ltsrec  27952  ltslpss  28059  cofcutr  28075  lenegs  28197  lesubsubsbd  28237  lesubsubs2bd  28238  lesubsubs3bd  28239  lesubaddsd  28244  lemuls2d  28325  lemuls1d  28326  ltonold  28412  oncutlt  28415  onnolt  28417  onles  28419  om2noseqlt2  28451  n0fincut  28506  bdaypw2n0bndlem  28614  bdaypw2bnd  28616  bdayfinbndlem1  28618  z12bdaylem1  28621