Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-bnj19 | Structured version Visualization version GIF version |
Description: Define the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
df-bnj19 | ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | cR | . . 3 class 𝑅 | |
4 | 1, 2, 3 | w-bnj19 32245 | . 2 wff TrFo(𝐵, 𝐴, 𝑅) |
5 | vx | . . . . . 6 setvar 𝑥 | |
6 | 5 | cv 1541 | . . . . 5 class 𝑥 |
7 | 1, 3, 6 | c-bnj14 32237 | . . . 4 class pred(𝑥, 𝐴, 𝑅) |
8 | 7, 2 | wss 3843 | . . 3 wff pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵 |
9 | 8, 5, 2 | wral 3053 | . 2 wff ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵 |
10 | 4, 9 | wb 209 | 1 wff ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵) |
Colors of variables: wff setvar class |
This definition is referenced by: bnj978 32500 bnj1118 32535 bnj1125 32543 bnj1137 32546 bnj1408 32587 |
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