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Mirrors > Home > MPE Home > Th. List > predexg | Structured version Visualization version GIF version |
Description: The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.) |
Ref | Expression |
---|---|
predexg | ⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6197 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | inex1g 5243 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V) | |
3 | 1, 2 | eqeltrid 2843 | 1 ⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3431 ∩ cin 3887 {csn 4563 ◡ccnv 5585 “ cima 5589 Predcpred 6196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3433 df-in 3895 df-pred 6197 |
This theorem is referenced by: predasetexOLD 6216 |
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