![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6254 | . 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋})) | |
2 | 0in 4354 | . 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅ | |
3 | 1, 2 | eqtri 2761 | 1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∩ cin 3910 ∅c0 4283 {csn 4587 ◡ccnv 5633 “ cima 5637 Predcpred 6253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-in 3918 df-nul 4284 df-pred 6254 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |