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Mirrors > Home > MPE Home > Th. List > predso | Structured version Visualization version GIF version |
Description: Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.) |
Ref | Expression |
---|---|
predso | ⊢ ((𝑅 Or 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sopo 5486 | . 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
2 | predpo 6160 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))) | |
3 | 1, 2 | sylan 582 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ⊆ wss 3935 Po wpo 5466 Or wor 5467 Predcpred 6141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-po 5468 df-so 5469 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 |
This theorem is referenced by: (None) |
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