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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 6150 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | inss1 4207 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴 | |
3 | 1, 2 | eqsstri 4003 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3937 ⊆ wss 3938 {csn 4569 ◡ccnv 5556 “ cima 5560 Predcpred 6149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-ss 3954 df-pred 6150 |
This theorem is referenced by: wfr3g 7955 wfrlem4 7960 wfrlem10 7966 trpredlem1 33068 wsuclem 33114 frr3g 33123 fpr3g 33124 frrlem4 33128 frrlem13 33137 fpr1 33141 |
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