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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 5597 | . 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋})) | |
2 | 0in 3921 | . 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅ | |
3 | 1, 2 | eqtri 2632 | 1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∩ cin 3539 ∅c0 3874 {csn 4125 ◡ccnv 5037 “ cima 5041 Predcpred 5596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-nul 3875 df-pred 5597 |
This theorem is referenced by: trpred0 30980 |
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