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Mirrors > Home > MPE Home > Th. List > Mathboxes > brredundsredund | Structured version Visualization version GIF version |
Description: For sets, binary relation on the class of all redundant sets (brredunds 38608) is equivalent to satisfying the redundancy predicate (df-redund 38606). (Contributed by Peter Mazsa, 25-Oct-2022.) |
Ref | Expression |
---|---|
brredundsredund | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds 〈𝐵, 𝐶〉 ↔ 𝐴 Redund 〈𝐵, 𝐶〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brredunds 38608 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))) | |
2 | df-redund 38606 | . 2 ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
3 | 1, 2 | bitr4di 289 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds 〈𝐵, 𝐶〉 ↔ 𝐴 Redund 〈𝐵, 𝐶〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 〈cop 4637 class class class wbr 5148 Redunds credunds 38182 Redund wredund 38183 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-oprab 7435 df-redunds 38605 df-redund 38606 |
This theorem is referenced by: (None) |
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