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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 5597 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | 1 | elin2 3763 | . 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
3 | elpredim.1 | . . . . 5 ⊢ 𝑋 ∈ V | |
4 | elimasng 5410 | . . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑋, 𝑌〉 ∈ ◡𝑅)) | |
5 | opelcnvg 5224 | . . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (〈𝑋, 𝑌〉 ∈ ◡𝑅 ↔ 〈𝑌, 𝑋〉 ∈ 𝑅)) | |
6 | 4, 5 | bitrd 267 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑌, 𝑋〉 ∈ 𝑅)) |
7 | 3, 6 | mpan 702 | . . . 4 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 〈𝑌, 𝑋〉 ∈ 𝑅)) |
8 | 7 | ibi 255 | . . 3 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → 〈𝑌, 𝑋〉 ∈ 𝑅) |
9 | df-br 4584 | . . 3 ⊢ (𝑌𝑅𝑋 ↔ 〈𝑌, 𝑋〉 ∈ 𝑅) | |
10 | 8, 9 | sylibr 223 | . 2 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → 𝑌𝑅𝑋) |
11 | 2, 10 | simplbiim 657 | 1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 “ cima 5041 Predcpred 5596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 |
This theorem is referenced by: predbrg 5617 preddowncl 5624 trpredrec 30982 |
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