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| Mirrors > Home > MPE Home > Th. List > Mathboxes > disjeq1i | Structured version Visualization version GIF version | ||
| Description: Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.) |
| Ref | Expression |
|---|---|
| disjeq1i.1 | ⊢ 𝐴 = 𝐵 |
| Ref | Expression |
|---|---|
| disjeq1i | ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | disjeq1i.1 | . . . 4 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | rmoeqi 36166 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) |
| 3 | 2 | albii 1819 | . 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) |
| 4 | df-disj 5109 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶) | |
| 5 | df-disj 5109 | . 2 ⊢ (Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wrmo 3378 Disj wdisj 5108 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2539 df-cleq 2728 df-clel 2815 df-rmo 3379 df-disj 5109 |
| This theorem is referenced by: disjeq12i 36172 |
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