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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 6291 | . 2 ⊢ Pred(𝑅, ∅, 𝑋) = (∅ ∩ (◡𝑅 “ {𝑋})) | |
2 | 0in 4386 | . 2 ⊢ (∅ ∩ (◡𝑅 “ {𝑋})) = ∅ | |
3 | 1, 2 | eqtri 2752 | 1 ⊢ Pred(𝑅, ∅, 𝑋) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∩ cin 3940 ∅c0 4315 {csn 4621 ◡ccnv 5666 “ cima 5670 Predcpred 6290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-in 3948 df-nul 4316 df-pred 6291 |
This theorem is referenced by: (None) |
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