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Mirrors > Home > ILE Home > Th. List > disjeq2 | GIF version |
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjeq2 | ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B ↔ Disj x ∈ A 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 2992 | . . . 4 ⊢ (B = 𝐶 → 𝐶 ⊆ B) | |
2 | 1 | ralimi 2378 | . . 3 ⊢ (∀x ∈ A B = 𝐶 → ∀x ∈ A 𝐶 ⊆ B) |
3 | disjss2 3739 | . . 3 ⊢ (∀x ∈ A 𝐶 ⊆ B → (Disj x ∈ A B → Disj x ∈ A 𝐶)) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B → Disj x ∈ A 𝐶)) |
5 | eqimss 2991 | . . . 4 ⊢ (B = 𝐶 → B ⊆ 𝐶) | |
6 | 5 | ralimi 2378 | . . 3 ⊢ (∀x ∈ A B = 𝐶 → ∀x ∈ A B ⊆ 𝐶) |
7 | disjss2 3739 | . . 3 ⊢ (∀x ∈ A B ⊆ 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B)) | |
8 | 6, 7 | syl 14 | . 2 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A 𝐶 → Disj x ∈ A B)) |
9 | 4, 8 | impbid 120 | 1 ⊢ (∀x ∈ A B = 𝐶 → (Disj x ∈ A B ↔ Disj x ∈ A 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∀wral 2300 ⊆ 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-ral 2305 df-rmo 2308 df-in 2918 df-ss 2925 df-disj 3737 |
This theorem is referenced by: disjeq2dv 3741 |
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