Theorem tz7.2 4389

Description: Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent _E Fr A. (Contributed by NM, 4-May-1994.)

Assertion

Ref	Expression
tz7.2	|- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) )

Proof of Theorem tz7.2

Step	Hyp	Ref	Expression
1		trss 4140	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
2		efrirr 4388	. . . . 5 |- ( _E Fr A -> -. A e. A )
3		eleq1 2259	. . . . . 6 |- ( B = A -> ( B e. A <-> A e. A ) )
4	3	notbid 668	. . . . 5 |- ( B = A -> ( -. B e. A <-> -. A e. A ) )
5	2, 4	syl5ibrcom 157	. . . 4 |- ( _E Fr A -> ( B = A -> -. B e. A ) )
6	5	necon2ad 2424	. . 3 |- ( _E Fr A -> ( B e. A -> B =/= A ) )
7	1, 6	anim12ii 343	. 2 |- ( ( Tr A /\ _E Fr A ) -> ( B e. A -> ( B C_ A /\ B =/= A ) ) )
8	7	3impia 1202	1 |- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   C_ wss 3157   Tr wtr 4131   _E cep 4322   Fr wfr 4363

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-tr 4132  df-eprel 4324  df-frfor 4366  df-frind 4367

This theorem is referenced by: (None)