Theorem slenlt 31877

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

Assertion

Ref	Expression
slenlt	|- ( ( A e. No /\ B e. No ) -> ( A ≤s B <-> -. B <s A ) )

Proof of Theorem slenlt

Step	Hyp	Ref	Expression
1		df-sle 31870	. . . 4 |- ≤s = ( ( No X. No ) \ `' <s )
2	1	breqi 4659	. . 3 |- ( A ≤s B <-> A ( ( No X. No ) \ `' <s ) B )
3		brdif 4705	. . 3 |- ( A ( ( No X. No ) \ `' <s ) B <-> ( A ( No X. No ) B /\ -. A `' <s B ) )
4		brxp 5147	. . . 4 |- ( A ( No X. No ) B <-> ( A e. No /\ B e. No ) )
5	4	anbi1i 731	. . 3 |- ( ( A ( No X. No ) B /\ -. A `' <s B ) <-> ( ( A e. No /\ B e. No ) /\ -. A `' <s B ) )
6	2, 3, 5	3bitri 286	. 2 |- ( A ≤s B <-> ( ( A e. No /\ B e. No ) /\ -. A `' <s B ) )
7		ibar 525	. . 3 |- ( ( A e. No /\ B e. No ) -> ( -. A `' <s B <-> ( ( A e. No /\ B e. No ) /\ -. A `' <s B ) ) )
8		brcnvg 5303	. . . 4 |- ( ( A e. No /\ B e. No ) -> ( A `' <s B <-> B <s A ) )
9	8	notbid 308	. . 3 |- ( ( A e. No /\ B e. No ) -> ( -. A `' <s B <-> -. B <s A ) )
10	7, 9	bitr3d 270	. 2 |- ( ( A e. No /\ B e. No ) -> ( ( ( A e. No /\ B e. No ) /\ -. A `' <s B ) <-> -. B <s A ) )
11	6, 10	syl5bb 272	1 |- ( ( A e. No /\ B e. No ) -> ( A ≤s B <-> -. B <s A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   \ cdif 3571   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   Nocsur 31793   <scslt 31794   ≤scsle 31869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-sle 31870

This theorem is referenced by:  sltnle  31878  sleloe  31879  sletri3  31880  sltletr  31881  slelttr  31882  sletr  31883  sltrec  31924