![]() |
Mathbox for Gino Giotto |
< Previous
Next >
Nearby theorems |
|
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 36129 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) |
3 | 2 | albii 1814 | . 2 ⊢ (∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐵 𝑡 ∈ 𝐶) |
4 | df-disj 5118 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑡∃*𝑥 ∈ 𝐴 𝑡 ∈ 𝐶) | |
5 | df-disj 5118 | . 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 1533 = wceq 1535 ∈ wcel 2104 ∃*wrmo 3375 Disj wdisj 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-mo 2536 df-cleq 2725 df-clel 2812 df-rmo 3376 df-disj 5118 |
This theorem is referenced by: disjeq12i 36135 |
Copyright terms: Public domain | W3C validator |