Theorem tz7.2 2183

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 ∈ A) → (B ⊆ A ∧ ¬ B = A))

Proof of Theorem tz7.2

Step	Hyp	Ref	Expression
1		trss 2050	. . . 4 ⊢ (Tr A → (B ∈ A → B ⊆ A))
2		eleq1 1149	. . . . . . . 8 ⊢ (B = A → (B ∈ A ↔ A ∈ A))
3	2	negbid 463	. . . . . . 7 ⊢ (B = A → (¬ B ∈ A ↔ ¬ A ∈ A))
4		efrirr 2180	. . . . . . 7 ⊢ (E Fr A → ¬ A ∈ A)
5	3, 4	syl5bir 184	. . . . . 6 ⊢ (B = A → (E Fr A → ¬ B ∈ A))
6	5	com12 13	. . . . 5 ⊢ (E Fr A → (B = A → ¬ B ∈ A))
7	6	con2d 83	. . . 4 ⊢ (E Fr A → (B ∈ A → ¬ B = A))
8	1, 7	anim12i 268	. . 3 ⊢ ((Tr A ∧ E Fr A) → ((B ∈ A → B ⊆ A) ∧ (B ∈ A → ¬ B = A)))
9		jcab 454	. . 3 ⊢ ((B ∈ A → (B ⊆ A ∧ ¬ B = A)) ↔ ((B ∈ A → B ⊆ A) ∧ (B ∈ A → ¬ B = A)))
10	8, 9	sylibr 175	. 2 ⊢ ((Tr A ∧ E Fr A) → (B ∈ A → (B ⊆ A ∧ ¬ B = A)))
11	10	imp 277	1 ⊢ (((Tr A ∧ E Fr A) ∧ B ∈ A) → (B ⊆ A ∧ ¬ B = A))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  Tr wtr 2041  Ecep 2056   Fr wfr 2061

This theorem is referenced by:  tz7.7 2224

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169

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