Theorem tz7.2 4372

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 4125	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
2		efrirr 4371	. . . . 5 |- ( _E Fr A -> -. A e. A )
3		eleq1 2252	. . . . . 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 2417	. . 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 2160   =/= wne 2360   C_ wss 3144   Tr wtr 4116   _E cep 4305   Fr wfr 4346

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 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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-tr 4117  df-eprel 4307  df-frfor 4349  df-frind 4350

This theorem is referenced by: (None)