Theorem tz7.2 4401

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 4151	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
2		efrirr 4400	. . . . 5 |- ( _E Fr A -> -. A e. A )
3		eleq1 2268	. . . . . 6 |- ( B = A -> ( B e. A <-> A e. A ) )
4	3	notbid 669	. . . . 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 2433	. . 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 1203	1 |- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176   =/= wne 2376   C_ wss 3166   Tr wtr 4142   _E cep 4334   Fr wfr 4375

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-tr 4143  df-eprel 4336  df-frfor 4378  df-frind 4379

This theorem is referenced by: (None)