Theorem predbrg 6136

Description: Closed form of elpredim 6128. (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 6120	. . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
2	1	eleq2d 2875	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
3		breq2 5034	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑌𝑅𝑥 ↔ 𝑌𝑅𝑋))
4	2, 3	imbi12d 348	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥) ↔ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)))
5		vex 3444	. . . 4 ⊢ 𝑥 ∈ V
6	5	elpredim 6128	. . 3 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥)
7	4, 6	vtoclg 3515	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋))
8	7	imp 410	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  Predcpred 6115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116

This theorem is referenced by: (None)