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Mirrors > Home > MPE Home > Th. List > disjeq2 | Structured version Visualization version 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 3972 | . . . 4 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵) | |
2 | 1 | ralimi 3128 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) |
3 | disjss2 4998 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 𝐶)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 𝐶)) |
5 | eqimss 3971 | . . . 4 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶) | |
6 | 5 | ralimi 3128 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶) |
7 | disjss2 4998 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) |
9 | 4, 8 | impbid 215 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∀wral 3106 ⊆ wss 3881 Disj wdisj 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-mo 2598 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-rmo 3114 df-v 3443 df-in 3888 df-ss 3898 df-disj 4996 |
This theorem is referenced by: disjeq2dv 5000 voliun 24158 carsgclctunlem2 31687 mblfinlem2 35095 voliunnfl 35101 |
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