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Mirrors > Home > MPE Home > Th. List > predel | Structured version Visualization version GIF version |
Description: Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.) |
Ref | Expression |
---|---|
predel | ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinel1 4174 | . 2 ⊢ (𝑌 ∈ (𝐴 ∩ (◡𝑅 “ {𝑋})) → 𝑌 ∈ 𝐴) | |
2 | df-pred 6150 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
3 | 1, 2 | eleq2s 2933 | 1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∩ cin 3937 {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-pred 6150 |
This theorem is referenced by: predpo 6168 predpoirr 6178 predfrirr 6179 dftrpred3g 33074 |
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