Theorem trelpss 40793

Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5542, ax-reg 9059 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 9071	. . 3 ⊢ E Fr 𝐴
2		tz7.2 5542	. . 3 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
3	1, 2	mp3an2 1445	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
4		df-pss 3957	. 2 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
5	3, 4	sylibr 236	1 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2113   ≠ wne 3019   ⊆ wss 3939   ⊊ wpss 3940  Tr wtr 5175   E cep 5467   Fr wfr 5514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-reg 9059

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-tr 5176  df-eprel 5468  df-fr 5517

This theorem is referenced by: (None)