Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > disjeq1 | Structured version Visualization version 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 4023 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | disjss1 5029 | . . 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 4022 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | disjss1 5029 | . . 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 214 | 1 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ⊆ wss 3935 Disj wdisj 5023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-clab 2800 df-cleq 2814 df-clel 2893 df-rmo 3146 df-in 3942 df-ss 3951 df-disj 5024 |
This theorem is referenced by: disjeq1d 5031 volfiniun 24142 disjrnmpt 30329 iundisj2cnt 30516 unelldsys 31412 sigapildsys 31416 ldgenpisyslem1 31417 rossros 31434 measvun 31463 pmeasmono 31577 pmeasadd 31578 meadjuni 42733 |
Copyright terms: Public domain | W3C validator |