![]() |
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 6299 | . 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋})) | |
2 | 0in 4389 | . 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅ | |
3 | 1, 2 | eqtri 2756 | 1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∩ cin 3944 ∅c0 4318 {csn 4624 ◡ccnv 5671 “ cima 5675 Predcpred 6298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-dif 3948 df-in 3952 df-nul 4319 df-pred 6299 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |