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Mirrors > Home > MPE Home > Th. List > trpred | Structured version Visualization version GIF version |
Description: The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.) |
Ref | Expression |
---|---|
trpred | ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | predep 6324 | . . 3 ⊢ (𝑋 ∈ 𝐴 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋)) | |
2 | 1 | adantl 481 | . 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋)) |
3 | trss 5269 | . . . 4 ⊢ (Tr 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ⊆ 𝐴)) | |
4 | 3 | imp 406 | . . 3 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ⊆ 𝐴) |
5 | sseqin2 4210 | . . 3 ⊢ (𝑋 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑋) = 𝑋) | |
6 | 4, 5 | sylib 217 | . 2 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∩ 𝑋) = 𝑋) |
7 | 2, 6 | eqtrd 2766 | 1 ⊢ ((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ⊆ wss 3943 Tr wtr 5258 E cep 5572 Predcpred 6292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 |
This theorem is referenced by: predon 7769 omsinds 7872 |
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