Theorem tz7.2 4451

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 4196	. . 3 |- ( Tr A -> ( B e. A -> B C_ A ) )
2		efrirr 4450	. . . . 5 |- ( _E Fr A -> -. A e. A )
3		eleq1 2294	. . . . . 6 |- ( B = A -> ( B e. A <-> A e. A ) )
4	3	notbid 673	. . . . 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 2459	. . 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 1226	1 |- ( ( Tr A /\ _E Fr A /\ B e. A ) -> ( B C_ A /\ B =/= A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   C_ wss 3200   Tr wtr 4187   _E cep 4384   Fr wfr 4425

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-tr 4188  df-eprel 4386  df-frfor 4428  df-frind 4429

This theorem is referenced by: (None)