Theorem trelpss 45375

Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5631, ax-reg 9564 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 9583	. . 3 ⊢ E Fr 𝐴
2		tz7.2 5631	. . 3 ⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
3	1, 2	mp3an2 1478	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
4		df-pss 3919	. 2 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
5	3, 4	sylibr 237	1 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊊ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ≠ wne 2955   ⊆ wss 3899   ⊊ wpss 3900  Tr wtr 5212   E cep 5547   Fr wfr 5598

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pr 5391  ax-reg 9564

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5548  df-fr 5601

This theorem is used by: (None)