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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 5841 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | inss1 3976 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴 | |
3 | 1, 2 | eqsstri 3776 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3714 ⊆ wss 3715 {csn 4321 ◡ccnv 5265 “ cima 5269 Predcpred 5840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-v 3342 df-in 3722 df-ss 3729 df-pred 5841 |
This theorem is referenced by: wfr3g 7583 wfrlem4 7588 wfrlem10 7594 trpredlem1 32053 wsuclem 32097 frr3g 32106 frrlem4 32110 |
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