Theorem trpred 6336

Description: The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.)

Assertion

Ref	Expression
trpred	⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)

Proof of Theorem trpred

Step	Hyp	Ref	Expression
1		predep 6335	. . 3 ⊢ (𝑋 ∈ 𝐴 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))
2	1	adantl 487	. 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))
3		trss 5221	. . . 4 ⊢ (Tr 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ⊆ 𝐴))
4	3	imp 412	. . 3 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ⊆ 𝐴)
5		sseqin2 4119	. . 3 ⊢ (𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑋) = 𝑋)
6	4, 5	sylib 221	. 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∩ 𝑋) = 𝑋)
7	2, 6	eqtrd 2795	1 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896   ∩ cin 4088  Tr wtr 5211   E cep 5549  Predcpred 6305

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  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-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306

This theorem is used by:  predon  7802  omsinds  7900  trfr  46059