Theorem tz7.2 4356

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 4112	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
2		efrirr 4355	. . . . 5 |- ( _E Fr A -> -. A e. A )
3		eleq1 2240	. . . . . 6 |- ( B = A -> ( B e. A <-> A e. A ) )
4	3	notbid 667	. . . . 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 2404	. . 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 1200	1 |- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   C_ wss 3131   Tr wtr 4103   _E cep 4289   Fr wfr 4330

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-tr 4104  df-eprel 4291  df-frfor 4333  df-frind 4334

This theorem is referenced by: (None)