Theorem trpred 6360

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 6359	. . 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 5228	. . . 4 ⊢ (Tr 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ⊆ 𝐴))
4	3	imp 412	. . 3 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ⊆ 𝐴)
5		sseqin2 4123	. . 3 ⊢ (𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑋) = 𝑋)
6	4, 5	sylib 221	. 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∩ 𝑋) = 𝑋)
7	2, 6	eqtrd 2798	1 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  Tr wtr 5218   E cep 5557  Predcpred 6329

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330

This theorem is used by:  predon  7834  omsinds  7932  trfr  46216