Theorem predbrg 6213

Description: Closed form of elpredim 6207. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)

Assertion

Ref	Expression
predbrg	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)

Proof of Theorem predbrg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		predeq3 6195	. . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2	1	eleq2d 2824	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
3		breq2 5074	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑌𝑅𝑥 ↔ 𝑌𝑅𝑋))
4	2, 3	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥) ↔ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)))
5		vex 3426	. . . 4 ⊢ 𝑥 ∈ V
6	5	elpredim 6207	. . 3 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥)
7	4, 6	vtoclg 3495	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋))
8	7	imp 406	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   class class class wbr 5070  Predcpred 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191

This theorem is referenced by:  predtrss  6214