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| 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 4042 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
| 2 | disjss1 5115 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶)) | 
| 4 | eqimss 4041 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
| 5 | disjss1 5115 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) | |
| 6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶)) | 
| 7 | 3, 6 | impbid 212 | 1 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ⊆ wss 3950 Disj wdisj 5109 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-mo 2539 df-cleq 2728 df-clel 2815 df-rmo 3379 df-ss 3967 df-disj 5110 | 
| This theorem is referenced by: disjeq1d 5117 volfiniun 25583 disjrnmpt 32599 iundisj2cnt 32802 unelldsys 34160 sigapildsys 34164 ldgenpisyslem1 34165 rossros 34182 measvun 34211 pmeasmono 34327 pmeasadd 34328 meadjuni 46477 | 
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