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Mirrors > Home > MPE Home > Th. List > elpredg | Structured version Visualization version GIF version |
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) (Proof shortened by BJ, 16-Oct-2024.) |
Ref | Expression |
---|---|
elpredg | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpredgg 6267 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | |
2 | ibar 530 | . . . 4 ⊢ (𝑌 ∈ 𝐴 → (𝑌𝑅𝑋 ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋))) | |
3 | 2 | bicomd 222 | . . 3 ⊢ (𝑌 ∈ 𝐴 → ((𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋)) |
4 | 3 | adantl 483 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → ((𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋) ↔ 𝑌𝑅𝑋)) |
5 | 1, 4 | bitrd 279 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5106 Predcpred 6253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 |
This theorem is referenced by: predpoirr 6288 predfrirr 6289 wfrlem10OLD 8265 wsuclem 34456 wsuclb 34459 |
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