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Mirrors > Home > MPE Home > Th. List > pred0 | Structured version Visualization version GIF version |
Description: The predecessor class over ∅ is always ∅. (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.) |
Ref | Expression |
---|---|
pred0 | ⊢ Pred(𝑅, ∅, 𝑋) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6252 | . 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋})) | |
2 | 0in 4352 | . 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅ | |
3 | 1, 2 | eqtri 2764 | 1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∩ cin 3908 ∅c0 4281 {csn 4585 ◡ccnv 5631 “ cima 5635 Predcpred 6251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3407 df-v 3446 df-dif 3912 df-in 3916 df-nul 4282 df-pred 6252 |
This theorem is referenced by: (None) |
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