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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 6254 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | inex1g 5277 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V) | |
3 | 1, 2 | eqeltrid 2838 | 1 ⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3444 ∩ cin 3910 {csn 4587 ◡ccnv 5633 “ cima 5637 Predcpred 6253 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-in 3918 df-pred 6254 |
This theorem is referenced by: predasetexOLD 6273 |
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