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Mirrors > Home > MPE Home > Th. List > Mathboxes > trpredpo | Structured version Visualization version GIF version |
Description: If 𝑅 partially orders 𝐴, then the transitive predecessors are the same as the immediate predecessors . (Contributed by Scott Fenton, 28-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
trpredpo | ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1133 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → 𝑋 ∈ 𝐴) | |
2 | simp3 1134 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → 𝑅 Se 𝐴) | |
3 | predpo 6166 | . . . . 5 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑋))) | |
4 | 3 | ralrimiv 3181 | . . . 4 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑋)Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑋)) |
5 | 4 | 3adant3 1128 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑋)Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑋)) |
6 | ssidd 3990 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋)) | |
7 | trpredmintr 33070 | . . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑋)Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑋) ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋))) → TrPred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋)) | |
8 | 1, 2, 5, 6, 7 | syl22anc 836 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐴, 𝑋)) |
9 | setlikespec 6169 | . . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V) | |
10 | trpredpred 33067 | . . . 4 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → Pred(𝑅, 𝐴, 𝑋) ⊆ TrPred(𝑅, 𝐴, 𝑋)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ⊆ TrPred(𝑅, 𝐴, 𝑋)) |
12 | 11 | 3adant1 1126 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ⊆ TrPred(𝑅, 𝐴, 𝑋)) |
13 | 8, 12 | eqssd 3984 | 1 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 Po wpo 5472 Se wse 5512 Predcpred 6147 TrPredctrpred 33056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-trpred 33057 |
This theorem is referenced by: (None) |
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