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Mirrors > Home > ILE Home > Th. List > disjeq1 | GIF version |
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjeq1 | ⊢ (A = B → (Disj x ∈ A 𝐶 ↔ Disj x ∈ B 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 2992 | . . 3 ⊢ (A = B → B ⊆ A) | |
2 | disjss1 3742 | . . 3 ⊢ (B ⊆ A → (Disj x ∈ A 𝐶 → Disj x ∈ B 𝐶)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (A = B → (Disj x ∈ A 𝐶 → Disj x ∈ B 𝐶)) |
4 | eqimss 2991 | . . 3 ⊢ (A = B → A ⊆ B) | |
5 | disjss1 3742 | . . 3 ⊢ (A ⊆ B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (A = B → (Disj x ∈ B 𝐶 → Disj x ∈ A 𝐶)) |
7 | 3, 6 | impbid 120 | 1 ⊢ (A = B → (Disj x ∈ A 𝐶 ↔ Disj x ∈ B 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ⊆ wss 2911 Disj wdisj 3736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-rmo 2308 df-in 2918 df-ss 2925 df-disj 3737 |
This theorem is referenced by: disjeq1d 3744 |
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