Theorem predbrg 5919

Description: Closed form of elpredim 5911. (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 5903	. . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2	1	eleq2d 2865	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
3		breq2 4848	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑌𝑅𝑥 ↔ 𝑌𝑅𝑋))
4	2, 3	imbi12d 336	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥) ↔ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)))
5		vex 3389	. . . 4 ⊢ 𝑥 ∈ V
6	5	elpredim 5911	. . 3 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥)
7	4, 6	vtoclg 3454	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋))
8	7	imp 396	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157   class class class wbr 4844  Predcpred 5898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-xp 5319  df-cnv 5321  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899

This theorem is referenced by: (None)