Theorem lenlts 28139

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 28132	. . . 4 ⊢ ≤s = ((No × No) ∖ ◡ <s )
2	1	breqi 5112	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴((No × No) ∖ ◡ <s )𝐵)
3		brdif 5161	. . 3 ⊢ (𝐴((No × No) ∖ ◡ <s )𝐵 ↔ (𝐴(No × No)𝐵 ∧ ¬ 𝐴◡ <s 𝐵))
4		brxp 5704	. . . 4 ⊢ (𝐴(No × No)𝐵 ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No))
5	4	anbi1i 636	. . 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 538	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (¬ 𝐴◡ <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No) ∧ ¬ 𝐴◡ <s 𝐵)))
8		brcnvg 5864	. . . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147   ∖ cdif 4048   class class class wbr 5106   × cxp 5653  ^◡ccnv 5654  Nocsur 28027   <s clts 28028   ≤s cles 28131

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-les 28132

This theorem is used by:  ltnles  28140  lesloe  28141  lestri3  28142  lesnltd  28143  ltlestr  28147  leltstr  28148  lestr  28149  lesid  28154  lestric  28155  ltlesd  28160  ltsrec  28217  ltslpss  28324  cofcutr  28340  lenegs  28462  lesubsubsbd  28502  lesubsubs2bd  28503  lesubsubs3bd  28504  lesubaddsd  28509  lemuls2d  28590  lemuls1d  28591  ltonold  28677  oncutlt  28680  onnolt  28682  onles  28684  om2noseqlt2  28716  n0fincut  28771  bdaypw2n0bndlem  28879  bdaypw2bnd  28881  bdayfinbndlem1  28883  z12bdaylem1  28886