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Mirrors > Home > MPE Home > Th. List > predbrg | Structured version Visualization version GIF version |
Description: Closed form of elpredim 6141. (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 6133 | . . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋)) | |
2 | 1 | eleq2d 2818 | . . . 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 3402 | . . . 4 ⊢ 𝑥 ∈ V | |
6 | 5 | elpredim 6141 | . . 3 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑥) → 𝑌𝑅𝑥) |
7 | 4, 6 | vtoclg 3470 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)) |
8 | 7 | imp 410 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 Predcpred 6128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-xp 5531 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 |
This theorem is referenced by: (None) |
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