Theorem wessep 4349

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 3003	. . . . . . 7 |- ( B C_ A -> ( x e. B -> x e. A ) )
2		ssel 3003	. . . . . . 7 |- ( B C_ A -> ( y e. B -> y e. A ) )
3		ssel 3003	. . . . . . 7 |- ( B C_ A -> ( z e. B -> z e. A ) )
4	1, 2, 3	3anim123d 1251	. . . . . 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 271	. . . . 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 434	. . . 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 4144	. . . . . 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 461	. . . . 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 4076	. . . . . 6 |- ( x _E y <-> x e. y )
10		epel 4076	. . . . . 6 |- ( y _E z <-> y e. z )
11	9, 10	anbi12i 448	. . . . 5 |- ( ( x _E y /\ y _E z ) <-> ( x e. y /\ y e. z ) )
12		epel 4076	. . . . 5 |- ( x _E z <-> x e. z )
13	8, 11, 12	3imtr4g 203	. . . 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 2449	. 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 4345	. . 3 |- _E Fr B
17		df-wetr 4118	. . 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 882	. 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 132	1 |- ( ( _E We A /\ B C_ A ) -> _E We B )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   e. wcel 1434   A.wral 2353   C_ wss 2983   class class class wbr 3806   _E cep 4071   Fr wfr 4112   We wwe 4114

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-setind 4309

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-eprel 4073  df-frfor 4115  df-frind 4116  df-wetr 4118

This theorem is referenced by: (None)