Theorem tz7.2 5667

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 5269	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
2		efrirr 5664	. . . . 5 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)
3		eleq1 2828	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
4	3	notbid 318	. . . . 5 ⊢ (𝐵 = 𝐴 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
5	2, 4	syl5ibrcom 247	. . . 4 ⊢ ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵 ∈ 𝐴))
6	5	necon2ad 2954	. . 3 ⊢ ( E Fr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ≠ 𝐴))
7	1, 6	anim12ii 618	. 2 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))
8	7	3impia 1117	1 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2939   ⊆ wss 3950  Tr wtr 5258   E cep 5582   Fr wfr 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-fr 5636

This theorem is referenced by:  tz7.7  6409  trelpss  44479