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Mirrors > Home > MPE Home > Th. List > predbrg | Structured version Visualization version GIF version |
Description: Closed form of elpredim 6309. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.) |
Ref | Expression |
---|---|
predbrg | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | predeq3 6297 | . . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋)) | |
2 | 1 | eleq2d 2813 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋))) |
3 | breq2 5145 | . . . 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 3472 | . . . 4 ⊢ 𝑥 ∈ V | |
6 | 5 | elpredim 6309 | . . 3 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥) |
7 | 4, 6 | vtoclg 3537 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)) |
8 | 7 | imp 406 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 Predcpred 6292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 |
This theorem is referenced by: predtrss 6316 |
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