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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 31595. 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 4846 | . 2 ⊢ (𝑦 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑌𝑅𝑋)) | |
2 | df-bnj14 31275 | . 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | |
3 | 1, 2 | elrab2 3560 | 1 ⊢ (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 ∈ wcel 2157 class class class wbr 4843 predc-bnj14 31274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-bnj14 31275 |
This theorem is referenced by: bnj1175 31589 bnj1177 31591 bnj1388 31618 |
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