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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1418 | Structured version Visualization version GIF version |
Description: Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1418 | ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5150 | . 2 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝑥 ↔ 𝑦𝑅𝑥)) | |
2 | df-bnj14 34681 | . . 3 ⊢ pred(𝑥, 𝐴, 𝑅) = {𝑧 ∈ 𝐴 ∣ 𝑧𝑅𝑥} | |
3 | 2 | bnj1538 34847 | . 2 ⊢ (𝑧 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑧𝑅𝑥) |
4 | 1, 3 | vtoclga 3576 | 1 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5147 predc-bnj14 34680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-bnj14 34681 |
This theorem is referenced by: bnj1417 35033 bnj1523 35063 |
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