Theorem tz7.2 4480

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. (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 4222	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
2		efrirr 4479	. . . . 5 ⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)
3		eleq1 2297	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
4	3	notbid 673	. . . . 5 ⊢ (𝐵 = 𝐴 → (¬ 𝐵 ∈ 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
5	2, 4	syl5ibrcom 157	. . . 4 ⊢ ( E Fr 𝐴 → (𝐵 = 𝐴 → ¬ 𝐵 ∈ 𝐴))
6	5	necon2ad 2471	. . 3 ⊢ ( E Fr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ≠ 𝐴))
7	1, 6	anim12ii 343	. 2 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → (𝐵 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴)))
8	7	3impia 1227	1 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   ≠ wne 2414   ⊆ wss 3214  Tr wtr 4213   E cep 4413   Fr wfr 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-tr 4214  df-eprel 4415  df-frfor 4457  df-frind 4458

This theorem is referenced by: (None)