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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.) |
Ref | Expression |
---|---|
elpredim.1 | ⊢ 𝑋 ∈ V |
Ref | Expression |
---|---|
elpredim | ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6141 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | 1 | elin2 4171 | . 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
3 | elpredim.1 | . . . . 5 ⊢ 𝑋 ∈ V | |
4 | elimasng 5948 | . . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ ◡𝑅)) | |
5 | opelcnvg 5744 | . . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (〈𝑋, 𝑌〉 ∈ ◡𝑅 ↔ 〈𝑌, 𝑋〉 ∈ 𝑅)) | |
6 | 4, 5 | bitrd 280 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑌, 𝑋〉 ∈ 𝑅)) |
7 | 3, 6 | mpan 686 | . . . 4 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑌, 𝑋〉 ∈ 𝑅)) |
8 | 7 | ibi 268 | . . 3 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → 〈𝑌, 𝑋〉 ∈ 𝑅) |
9 | df-br 5058 | . . 3 ⊢ (𝑌𝑅𝑋 ↔ 〈𝑌, 𝑋〉 ∈ 𝑅) | |
10 | 8, 9 | sylibr 235 | . 2 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → 𝑌𝑅𝑋) |
11 | 2, 10 | simplbiim 505 | 1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 class class class wbr 5057 ◡ccnv 5547 “ cima 5551 Predcpred 6140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 |
This theorem is referenced by: predbrg 6161 preddowncl 6168 trpredrec 32974 |
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