Theorem trelpss 43214

Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5661, ax-reg 9587 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)

Assertion

Ref	Expression
trelpss	⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴)

Proof of Theorem trelpss

Step	Hyp	Ref	Expression
1		zfregfr 9600	. . 3 ⊢ E Fr 𝐴
2		tz7.2 5661	. . 3 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
3	1, 2	mp3an2 1450	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
4		df-pss 3968	. 2 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
5	3, 4	sylibr 233	1 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107   ≠ wne 2941   ⊆ wss 3949   ⊊ wpss 3950  Tr wtr 5266   E cep 5580   Fr wfr 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-reg 9587

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-fr 5632

This theorem is referenced by: (None)