Theorem tz7.2 2931

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 2689	. . 3 |- (Tr A -> (B e. A -> B (_ A))
2		eleq1 1534	. . . . . 6 |- (B = A -> (B e. A <-> A e. A))
3	2	negbid 611	. . . . 5 |- (B = A -> (-. B e. A <-> -. A e. A))
4		efrirr 2928	. . . . 5 |- (E Fr A -> -. A e. A)
5	3, 4	syl5cbir 211	. . . 4 |- (E Fr A -> (B = A -> -. B e. A))
6	5	necon2ad 1614	. . 3 |- (E Fr A -> (B e. A -> B =/= A))
7	1, 6	anim12ii 559	. 2 |- ((Tr A /\ E Fr A) -> (B e. A -> (B (_ A /\ B =/= A)))
8	7	3impia 830	1 |- ((Tr A /\ E Fr A /\ B e. A) -> (B (_ A /\ B =/= A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585   (_ wss 2047  Tr wtr 2680  Ecep 2830   Fr wfr 2915

This theorem is referenced by:  tz7.7 2973

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-fr 2917

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