Theorem tz7.2 5646

Description: Similar to Theorem 7.2 of [TakeutiZaring] p. 35, except that the Axiom of Regularity is not required due to the antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.)

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 5230	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
2		efrirr 5643	. . . . 5 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)
3		eleq1 2853	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
4	3	notbid 321	. . . . 5 ⊢ (𝐵 = 𝐴 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
5	2, 4	syl5ibrcom 250	. . . 4 ⊢ ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵 ∈ 𝐴))
6	5	necon2ad 2975	. . 3 ⊢ ( E Fr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ≠ 𝐴))
7	1, 6	anim12ii 630	. 2 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))
8	7	3impia 1135	1 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ⊆ wss 3906  Tr wtr 5220   E cep 5562   Fr wfr 5613

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-fr 5616

This theorem is used by:  tz7.7  6390  trelpss  45221