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Mirrors > Home > MPE Home > Th. List > elpredim | Structured version Visualization version GIF version |
Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) (Proof shortened by BJ, 16-Oct-2024.) |
Ref | Expression |
---|---|
elpredim.1 | ⊢ 𝑋 ∈ V |
Ref | Expression |
---|---|
elpredim | ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpredim.1 | . 2 ⊢ 𝑋 ∈ V | |
2 | elpredimg 6325 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3473 class class class wbr 5152 Predcpred 6309 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 |
This theorem is referenced by: predbrg 6332 preddowncl 6343 |
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