Theorem brsslt 31900

Description: Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
brsslt	|- ( A < <s B <-> ( ( A e. _V /\ B e. _V ) /\ ( A C_ No /\ B C_ No /\ A. x e. A A. y e. B x <s y ) ) )

Distinct variable groups:   x, A, y   x, B, y

Proof of Theorem brsslt

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sslt 31897	. . 3 |- < <s = { <. a , b >. | ( a C_ No /\ b C_ No /\ A. x e. a A. y e. b x <s y ) }
2	1	bropaex12 5192	. 2 |- ( A < <s B -> ( A e. _V /\ B e. _V ) )
3		sseq1 3626	. . . 4 |- ( a = A -> ( a C_ No <-> A C_ No ) )
4		raleq 3138	. . . 4 |- ( a = A -> ( A. x e. a A. y e. b x <s y <-> A. x e. A A. y e. b x <s y ) )
5	3, 4	3anbi13d 1401	. . 3 |- ( a = A -> ( ( a C_ No /\ b C_ No /\ A. x e. a A. y e. b x <s y ) <-> ( A C_ No /\ b C_ No /\ A. x e. A A. y e. b x <s y ) ) )
6		sseq1 3626	. . . 4 |- ( b = B -> ( b C_ No <-> B C_ No ) )
7		raleq 3138	. . . . 5 |- ( b = B -> ( A. y e. b x <s y <-> A. y e. B x <s y ) )
8	7	ralbidv 2986	. . . 4 |- ( b = B -> ( A. x e. A A. y e. b x <s y <-> A. x e. A A. y e. B x <s y ) )
9	6, 8	3anbi23d 1402	. . 3 |- ( b = B -> ( ( A C_ No /\ b C_ No /\ A. x e. A A. y e. b x <s y ) <-> ( A C_ No /\ B C_ No /\ A. x e. A A. y e. B x <s y ) ) )
10	5, 9, 1	brabg 4994	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A < <s B <-> ( A C_ No /\ B C_ No /\ A. x e. A A. y e. B x <s y ) ) )
11	2, 10	biadan2 674	1 |- ( A < <s B <-> ( ( A e. _V /\ B e. _V ) /\ ( A C_ No /\ B C_ No /\ A. x e. A A. y e. B x <s y ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Nocsur 31793   <scslt 31794   < <scsslt 31896

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-sslt 31897

This theorem is referenced by:  ssltex1  31901  ssltex2  31902  ssltss1  31903  ssltss2  31904  ssltsep  31905  sssslt1  31906  sssslt2  31907  nulsslt  31908  nulssgt  31909  conway  31910  sslttr  31914  ssltun1  31915  ssltun2  31916  etasslt  31920  slerec  31923