Theorem trpred 6278

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 6277	. . 3 ⊢ (𝑋 ∈ 𝐴 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))
2	1	adantl 481	. 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))
3		trss 5208	. . . 4 ⊢ (Tr 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ⊆ 𝐴))
4	3	imp 406	. . 3 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ⊆ 𝐴)
5		sseqin2 4173	. . 3 ⊢ (𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑋) = 𝑋)
6	4, 5	sylib 218	. 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∩ 𝑋) = 𝑋)
7	2, 6	eqtrd 2766	1 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∩ cin 3901   ⊆ wss 3902  Tr wtr 5198   E cep 5515  Predcpred 6247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248

This theorem is referenced by:  predon  7719  omsinds  7817  trfr  44994