Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elpredgg | Structured version Visualization version GIF version |
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.) |
Ref | Expression |
---|---|
elpredgg | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pred 6149 | . . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) | |
2 | 1 | elin2 4101 | . 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋}))) |
3 | elinisegg 5950 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋)) | |
4 | 3 | anbi2d 632 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ((𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
5 | 2, 4 | syl5bb 286 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 {csn 4531 class class class wbr 5043 ◡ccnv 5539 “ cima 5543 Predcpred 6148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 |
This theorem is referenced by: elpredg 6162 elpredimg 6163 elpred 6165 |
Copyright terms: Public domain | W3C validator |