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Mirrors > Home > MPE Home > Th. List > disjors | Structured version Visualization version GIF version |
Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjors | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2974 | . . 3 ⊢ Ⅎ𝑖𝐵 | |
2 | nfcsb1v 3904 | . . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
3 | csbeq1a 3894 | . . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
4 | 1, 2, 3 | cbvdisj 5032 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵) |
5 | csbeq1 3883 | . . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
6 | 5 | disjor 5037 | . 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
7 | 4, 6 | bitri 276 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∨ wo 841 = wceq 1528 ∀wral 3135 ⦋csb 3880 ∩ cin 3932 ∅c0 4288 Disj wdisj 5022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-in 3940 df-nul 4289 df-disj 5023 |
This theorem is referenced by: disji2 5039 disjprgw 5052 disjprg 5053 disjxiun 5054 disjxun 5055 iundisj2 24077 disji2f 30255 disjpreima 30262 disjxpin 30266 iundisj2f 30268 disjunsn 30272 iundisj2fi 30446 disjxp1 41208 disjinfi 41330 |
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