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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 5680 | . 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | predasetex.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | 2 | inex1 4799 | . 2 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ∈ V |
4 | 1, 3 | eqeltri 2697 | 1 ⊢ Pred(𝑅, 𝐴, 𝑋) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 {csn 4177 ◡ccnv 5113 “ cima 5117 Predcpred 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-pred 5680 |
This theorem is referenced by: (None) |
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