Theorem trpred 6320

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 6319	. . 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 5240	. . . 4 ⊢ (Tr 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ⊆ 𝐴))
4	3	imp 406	. . 3 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ⊆ 𝐴)
5		sseqin2 4198	. . 3 ⊢ (𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑋) = 𝑋)
6	4, 5	sylib 218	. 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∩ 𝑋) = 𝑋)
7	2, 6	eqtrd 2770	1 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∩ cin 3925   ⊆ wss 3926  Tr wtr 5229   E cep 5552  Predcpred 6289

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-eprel 5553  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290

This theorem is referenced by:  predon  7780  omsinds  7882  trfr  44987