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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 4142 | . 2 ⊢ (𝑌 ∈ (𝐴 ∩ (◡𝑅 “ {𝑋})) → 𝑌 ∈ 𝐴) | |
2 | df-pred 6238 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
3 | 1, 2 | eleq2s 2855 | 1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∩ cin 3897 {csn 4573 ◡ccnv 5619 “ cima 5623 Predcpred 6237 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-in 3905 df-pred 6238 |
This theorem is referenced by: predtrss 6261 predpoirr 6272 predfrirr 6273 xpord2pred 34074 |
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