Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj93 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj97 30936. 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 30759 | . . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | |
2 | 1 | simprbi 480 | . . 3 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴) |
3 | df-bnj13 30757 | . . 3 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) | |
4 | 2, 3 | sylib 208 | . 2 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) |
5 | 4 | r19.21bi 2932 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 Fr wfr 5070 predc-bnj14 30754 Se w-bnj13 30756 FrSe w-bnj15 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-ral 2917 df-bnj13 30757 df-bnj15 30759 |
This theorem is referenced by: bnj96 30935 bnj97 30936 bnj149 30945 bnj150 30946 bnj518 30956 bnj1148 31064 |
Copyright terms: Public domain | W3C validator |