Theorem slenlt 33941

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 33934	. . . 4 ⊢ ≤s = (( No × No ) ∖ ◡ <s )
2	1	breqi 5080	. . 3 ⊢ (𝐴 ≤s 𝐵 ↔ 𝐴(( No × No ) ∖ ◡ <s )𝐵)
3		brdif 5127	. . 3 ⊢ (𝐴(( No × No ) ∖ ◡ <s )𝐵 ↔ (𝐴( No × No )𝐵 ∧ ¬ 𝐴◡ <s 𝐵))
4		brxp 5632	. . . 4 ⊢ (𝐴( No × No )𝐵 ↔ (𝐴 ∈ No ∧ 𝐵 ∈ No ))
5	4	anbi1i 624	. . 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 529	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (¬ 𝐴◡ <s 𝐵 ↔ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴◡ <s 𝐵)))
8		brcnvg 5782	. . . 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 280	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ¬ 𝐴◡ <s 𝐵) ↔ ¬ 𝐵 <s 𝐴))
11	6, 10	syl5bb 283	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ∖ cdif 3884   class class class wbr 5074   × cxp 5583  ^◡ccnv 5584   No csur 33829   <s cslt 33830   ≤s csle 33933

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pr 5351

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-br 5075  df-opab 5137  df-xp 5591  df-cnv 5593  df-sle 33934

This theorem is referenced by:  sltnle  33942  sleloe  33943  sletri3  33944  sltletr  33945  slelttr  33946  sletr  33947  slerflex  33952  sltrec  34000  sltlpss  34073  cofcutr  34078