Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > predss | Structured version Visualization version GIF version |
Description: The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
predss | ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 5597 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | inss1 3795 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴 | |
3 | 1, 2 | eqsstri 3598 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3539 ⊆ wss 3540 {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-in 3547 df-ss 3554 df-pred 5597 |
This theorem is referenced by: wfr3g 7300 wfrlem4 7305 wfrlem10 7311 trpredlem1 30971 wsuclem 31017 wsuclemOLD 31018 frr3g 31023 frrlem4 31027 |
Copyright terms: Public domain | W3C validator |