Theorem sssslt2 31907

Description: Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.)

Assertion

Ref	Expression
sssslt2	|- ( ( A < <s B /\ C C_ B ) -> A < <s C )

Proof of Theorem sssslt2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssltex1 31901	. . . 4 |- ( A < <s B -> A e. _V )
2	1	adantr 481	. . 3 |- ( ( A < <s B /\ C C_ B ) -> A e. _V )
3		ssltex2 31902	. . . . 5 |- ( A < <s B -> B e. _V )
4	3	adantr 481	. . . 4 |- ( ( A < <s B /\ C C_ B ) -> B e. _V )
5		simpr 477	. . . 4 |- ( ( A < <s B /\ C C_ B ) -> C C_ B )
6	4, 5	ssexd 4805	. . 3 |- ( ( A < <s B /\ C C_ B ) -> C e. _V )
7	2, 6	jca 554	. 2 |- ( ( A < <s B /\ C C_ B ) -> ( A e. _V /\ C e. _V ) )
8		ssltss1 31903	. . . 4 |- ( A < <s B -> A C_ No )
9	8	adantr 481	. . 3 |- ( ( A < <s B /\ C C_ B ) -> A C_ No )
10		ssltss2 31904	. . . . 5 |- ( A < <s B -> B C_ No )
11	10	adantr 481	. . . 4 |- ( ( A < <s B /\ C C_ B ) -> B C_ No )
12	5, 11	sstrd 3613	. . 3 |- ( ( A < <s B /\ C C_ B ) -> C C_ No )
13		ssltsep 31905	. . . . 5 |- ( A < <s B -> A. x e. A A. y e. B x <s y )
14	13	adantr 481	. . . 4 |- ( ( A < <s B /\ C C_ B ) -> A. x e. A A. y e. B x <s y )
15		ssralv 3666	. . . . . 6 |- ( C C_ B -> ( A. y e. B x <s y -> A. y e. C x <s y ) )
16	5, 15	syl 17	. . . . 5 |- ( ( A < <s B /\ C C_ B ) -> ( A. y e. B x <s y -> A. y e. C x <s y ) )
17	16	ralimdv 2963	. . . 4 |- ( ( A < <s B /\ C C_ B ) -> ( A. x e. A A. y e. B x <s y -> A. x e. A A. y e. C x <s y ) )
18	14, 17	mpd 15	. . 3 |- ( ( A < <s B /\ C C_ B ) -> A. x e. A A. y e. C x <s y )
19	9, 12, 18	3jca 1242	. 2 |- ( ( A < <s B /\ C C_ B ) -> ( A C_ No /\ C C_ No /\ A. x e. A A. y e. C x <s y ) )
20		brsslt 31900	. 2 |- ( A < <s C <-> ( ( A e. _V /\ C e. _V ) /\ ( A C_ No /\ C C_ No /\ A. x e. A A. y e. C x <s y ) ) )
21	7, 19, 20	sylanbrc 698	1 |- ( ( A < <s B /\ C C_ B ) -> A < <s C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   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:  scutun12  31917