Theorem wessep 4705

Description: A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)

Assertion

Ref	Expression
wessep	|- ( ( _E We A /\ B C_ A ) -> _E We B )

Proof of Theorem wessep

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

Step	Hyp	Ref	Expression
1		ssel 3236	. . . . . . 7 |- ( B C_ A -> ( x e. B -> x e. A ) )
2		ssel 3236	. . . . . . 7 |- ( B C_ A -> ( y e. B -> y e. A ) )
3		ssel 3236	. . . . . . 7 |- ( B C_ A -> ( z e. B -> z e. A ) )
4	1, 2, 3	3anim123d 1356	. . . . . 6 |- ( B C_ A -> ( ( x e. B /\ y e. B /\ z e. B ) -> ( x e. A /\ y e. A /\ z e. A ) ) )
5	4	adantl 277	. . . . 5 |- ( ( _E We A /\ B C_ A ) -> ( ( x e. B /\ y e. B /\ z e. B ) -> ( x e. A /\ y e. A /\ z e. A ) ) )
6	5	imdistani 445	. . . 4 |- ( ( ( _E We A /\ B C_ A ) /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( _E We A /\ B C_ A ) /\ ( x e. A /\ y e. A /\ z e. A ) ) )
7		wetrep 4486	. . . . . 6 |- ( ( _E We A /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x e. y /\ y e. z ) -> x e. z ) )
8	7	adantlr 477	. . . . 5 |- ( ( ( _E We A /\ B C_ A ) /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x e. y /\ y e. z ) -> x e. z ) )
9		epel 4418	. . . . . 6 |- ( x _E y <-> x e. y )
10		epel 4418	. . . . . 6 |- ( y _E z <-> y e. z )
11	9, 10	anbi12i 460	. . . . 5 |- ( ( x _E y /\ y _E z ) <-> ( x e. y /\ y e. z ) )
12		epel 4418	. . . . 5 |- ( x _E z <-> x e. z )
13	8, 11, 12	3imtr4g 205	. . . 4 |- ( ( ( _E We A /\ B C_ A ) /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x _E y /\ y _E z ) -> x _E z ) )
14	6, 13	syl 14	. . 3 |- ( ( ( _E We A /\ B C_ A ) /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x _E y /\ y _E z ) -> x _E z ) )
15	14	ralrimivvva 2627	. 2 |- ( ( _E We A /\ B C_ A ) -> A. x e. B A. y e. B A. z e. B ( ( x _E y /\ y _E z ) -> x _E z ) )
16		zfregfr 4701	. . 3 |- _E Fr B
17		df-wetr 4460	. . 3 |- ( _E We B <-> ( _E Fr B /\ A. x e. B A. y e. B A. z e. B ( ( x _E y /\ y _E z ) -> x _E z ) ) )
18	16, 17	mpbiran 949	. 2 |- ( _E We B <-> A. x e. B A. y e. B A. z e. B ( ( x _E y /\ y _E z ) -> x _E z ) )
19	15, 18	sylibr 134	1 |- ( ( _E We A /\ B C_ A ) -> _E We B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   e. wcel 2205   A.wral 2522   C_ wss 3214   class class class wbr 4114   _E cep 4413   Fr wfr 4454   We wwe 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-eprel 4415  df-frfor 4457  df-frind 4458  df-wetr 4460

This theorem is referenced by: (None)