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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 2901 | . . 3 ⊢ Ⅎ𝑖𝐵 | |
2 | nfcsb1v 3917 | . . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
3 | csbeq1a 3906 | . . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
4 | 1, 2, 3 | cbvdisj 5122 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵) |
5 | csbeq1 3895 | . . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
6 | 5 | disjor 5127 | . 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
7 | 4, 6 | bitri 274 | 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 205 ∨ wo 843 = wceq 1539 ∀wral 3059 ⦋csb 3892 ∩ cin 3946 ∅c0 4321 Disj wdisj 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rmo 3374 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-in 3954 df-nul 4322 df-disj 5113 |
This theorem is referenced by: disji2 5129 disjprgw 5142 disjprg 5143 disjxiun 5144 disjxun 5145 iundisj2 25298 disji2f 32075 disjpreima 32082 disjxpin 32086 iundisj2f 32088 disjunsn 32092 iundisj2fi 32275 disjxp1 44057 disjinfi 44189 |
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