Theorem tz7.2 2926

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.

Assertion

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

Proof of Theorem tz7.2

Step	Hyp	Ref	Expression
1		trss 2684	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2		eleq1 1531	. . . . . 6 |- (B = A -> (B e. A <-> A e. A))
3	2	negbid 610	. . . . 5 |- (B = A -> (-. B e. A <-> -. A e. A))
4		efrirr 2923	. . . . 5 |- (E Fr A -> -. A e. A)
5	3, 4	syl5cbir 211	. . . 4 |- (E Fr A -> (B = A -> -. B e. A))
6	5	necon2ad 1611	. . 3 |- (E Fr A -> (B e. A -> B =/= A))
7	1, 6	anim12ii 558	. 2 |- ((Tr A /\ E Fr A) -> (B e. A -> (B (_ A /\ B =/= A)))
8	7	3impia 829	1 |- ((Tr A /\ E Fr A /\ B e. A) -> (B (_ A /\ B =/= A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   =/= wne 1582   (_ wss 2043  Tr wtr 2675  Ecep 2825   Fr wfr 2910

This theorem is referenced by:  tz7.7 2968

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-fr 2912

Copyright terms: Public domain