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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj93 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj97 32133. 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 31958 | . . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | |
2 | 1 | simprbi 499 | . . 3 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴) |
3 | df-bnj13 31956 | . . 3 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) | |
4 | 2, 3 | sylib 220 | . 2 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) |
5 | 4 | r19.21bi 3208 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 Fr wfr 5505 predc-bnj14 31953 Se w-bnj13 31955 FrSe w-bnj15 31957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-12 2173 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-ral 3143 df-bnj13 31956 df-bnj15 31958 |
This theorem is referenced by: bnj96 32132 bnj97 32133 bnj149 32142 bnj150 32143 bnj518 32153 bnj1148 32263 |
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