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| Mirrors > Home > MPE Home > Th. List > elpred | Structured version Visualization version GIF version | ||
| Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) (Proof shortened by BJ, 16-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| elpred.1 | ⊢ 𝑌 ∈ V | 
| Ref | Expression | 
|---|---|
| elpred | ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elpred.1 | . 2 ⊢ 𝑌 ∈ V | |
| 2 | elpredgg 6333 | . 2 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ V) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | |
| 3 | 1, 2 | mpan2 691 | 1 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 Vcvv 3479 class class class wbr 5142 Predcpred 6319 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 | 
| This theorem is referenced by: predtrss 6342 setlikespec 6345 preddowncl 6352 xpord2pred 8171 xpord3pred 8178 fprlem2 8327 wfrlem10OLD 8359 ttrclselem2 9767 ttrclse 9768 preduz 13691 predfz 13694 wzel 35826 wsuclem 35827 | 
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