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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 6293 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | inex1g 5312 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V) | |
3 | 1, 2 | eqeltrid 2831 | 1 ⊢ (𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3468 ∩ cin 3942 {csn 4623 ◡ccnv 5668 “ cima 5672 Predcpred 6292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-in 3950 df-pred 6293 |
This theorem is referenced by: predasetexOLD 6312 |
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