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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1152 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 33622. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1152 | ⊢ (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5108 | . 2 ⊢ (𝑦 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑌𝑅𝑋)) | |
2 | df-bnj14 33301 | . 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | |
3 | 1, 2 | elrab2 3648 | 1 ⊢ (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5105 predc-bnj14 33300 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-bnj14 33301 |
This theorem is referenced by: bnj1175 33616 bnj1177 33618 bnj1388 33645 |
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