Theorem wessep 4611

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 3174	. . . . . . 7 |- ( B C_ A -> ( x e. B -> x e. A ) )
2		ssel 3174	. . . . . . 7 |- ( B C_ A -> ( y e. B -> y e. A ) )
3		ssel 3174	. . . . . . 7 |- ( B C_ A -> ( z e. B -> z e. A ) )
4	1, 2, 3	3anim123d 1330	. . . . . 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 4392	. . . . . 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 4324	. . . . . 6 |- ( x _E y <-> x e. y )
10		epel 4324	. . . . . 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 4324	. . . . 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 2577	. 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 4607	. . 3 |- _E Fr B
17		df-wetr 4366	. . 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 942	. 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 980   e. wcel 2164   A.wral 2472   C_ wss 3154   class class class wbr 4030   _E cep 4319   Fr wfr 4360   We wwe 4362

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-eprel 4321  df-frfor 4363  df-frind 4364  df-wetr 4366

This theorem is referenced by: (None)