Theorem lenlts 27716

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 27709	. . . 4 ⊢ ≤s = (( No × No ) ∖ ◡ <s )
2	1	breqi 5091	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ◡ <s )𝐵)
3		brdif 5138	. . 3 ⊢ (𝐴(( No × No ) ∖ ◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴◡ <s 𝐵))
4		brxp 5680	. . . 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 528	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴◡ <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴◡ <s 𝐵)))
8		brcnvg 5834	. . . 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))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ∖ cdif 3886   class class class wbr 5085   × cxp 5629  ^◡ccnv 5630   No csur 27603   <s clts 27604   ≤s cles 27708

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-les 27709

This theorem is referenced by:  ltnles  27717  lesloe  27718  lestri3  27719  lesnltd  27720  ltlestr  27724  leltstr  27725  lestr  27726  lesid  27731  lestric  27732  ltlesd  27737  ltsrec  27793  ltslpss  27900  cofcutr  27916  lenegs  28038  lesubsubsbd  28078  lesubsubs2bd  28079  lesubsubs3bd  28080  lesubaddsd  28085  lemuls2d  28166  lemuls1d  28167  ltonold  28253  oncutlt  28256  onnolt  28258  onles  28260  om2noseqlt2  28292  n0fincut  28347  bdaypw2n0bndlem  28455  bdaypw2bnd  28457  bdayfinbndlem1  28459  z12bdaylem1  28462