Theorem wessep 4487

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 3086	. . . . . . 7 |- ( B C_ A -> ( x e. B -> x e. A ) )
2		ssel 3086	. . . . . . 7 |- ( B C_ A -> ( y e. B -> y e. A ) )
3		ssel 3086	. . . . . . 7 |- ( B C_ A -> ( z e. B -> z e. A ) )
4	1, 2, 3	3anim123d 1297	. . . . . 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 275	. . . . 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 441	. . . 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 4277	. . . . . 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 468	. . . . 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 4209	. . . . . 6 |- ( x _E y <-> x e. y )
10		epel 4209	. . . . . 6 |- ( y _E z <-> y e. z )
11	9, 10	anbi12i 455	. . . . 5 |- ( ( x _E y /\ y _E z ) <-> ( x e. y /\ y e. z ) )
12		epel 4209	. . . . 5 |- ( x _E z <-> x e. z )
13	8, 11, 12	3imtr4g 204	. . . 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 2513	. 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 4483	. . 3 |- _E Fr B
17		df-wetr 4251	. . 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 924	. 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 133	1 |- ( ( _E We A /\ B C_ A ) -> _E We B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   e. wcel 1480   A.wral 2414   C_ wss 3066   class class class wbr 3924   _E cep 4204   Fr wfr 4245   We wwe 4247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-eprel 4206  df-frfor 4248  df-frind 4249  df-wetr 4251

This theorem is referenced by: (None)