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Mirrors > Home > MPE Home > Th. List > disjss2 | Structured version Visualization version GIF version |
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjss2 | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3908 | . . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) | |
2 | 1 | ralimi 3128 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
3 | rmoim 3679 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
5 | 4 | alimdv 1917 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)) |
6 | df-disj 4996 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
7 | df-disj 4996 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
8 | 5, 6, 7 | 3imtr4g 299 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∈ wcel 2111 ∀wral 3106 ∃*wrmo 3109 ⊆ 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: disjeq2 4999 0disj 5022 uniioombllem2 24187 uniioombllem4 24190 disjxwwlksn 27690 disjxwwlkn 27699 fusgreghash2wspv 28120 fsumiunss 42217 |
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