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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 35895) is equivalent to satisfying the redundancy predicate (df-redund 35893). (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 35895 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))) | |
2 | df-redund 35893 | . 2 ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
3 | 1, 2 | syl6bbr 291 | 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 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∩ cin 3928 ⊆ wss 3929 〈cop 4566 class class class wbr 5059 Redunds credunds 35507 Redund wredund 35508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-rel 5555 df-cnv 5556 df-oprab 7153 df-redunds 35892 df-redund 35893 |
This theorem is referenced by: (None) |
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