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Mirrors > Home > MPE Home > Th. List > predasetex | Structured version Visualization version GIF version |
Description: The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) |
Ref | Expression |
---|---|
predasetex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
predasetex | ⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6151 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | predasetex.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | inex1 5224 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V |
4 | 1, 3 | eqeltri 2912 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 Vcvv 3497 ∩ cin 3938 {csn 4570 ◡ccnv 5557 “ cima 5561 Predcpred 6150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-in 3946 df-pred 6151 |
This theorem is referenced by: (None) |
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