| Mathbox for Jonathan Ben-Naim |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj93 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj97 34880. 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 |
|---|---|
| bnj93 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-bnj15 34707 | . . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | |
| 2 | 1 | simprbi 496 | . . 3 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴) |
| 3 | df-bnj13 34705 | . . 3 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) | |
| 4 | 2, 3 | sylib 218 | . 2 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) |
| 5 | 4 | r19.21bi 3251 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 Fr wfr 5634 predc-bnj14 34702 Se w-bnj13 34704 FrSe w-bnj15 34706 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-ral 3062 df-bnj13 34705 df-bnj15 34707 |
| This theorem is referenced by: bnj96 34879 bnj97 34880 bnj149 34889 bnj150 34890 bnj518 34900 bnj1148 35010 |
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