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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 2904 | . . 3 ⊢ Ⅎ𝑖𝐵 | |
2 | nfcsb1v 3919 | . . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
3 | csbeq1a 3908 | . . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
4 | 1, 2, 3 | cbvdisj 5124 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵) |
5 | csbeq1 3897 | . . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
6 | 5 | disjor 5129 | . 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
7 | 4, 6 | bitri 275 | 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 846 = wceq 1542 ∀wral 3062 ⦋csb 3894 ∩ cin 3948 ∅c0 4323 Disj wdisj 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rmo 3377 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-in 3956 df-nul 4324 df-disj 5115 |
This theorem is referenced by: disji2 5131 disjprgw 5144 disjprg 5145 disjxiun 5146 disjxun 5147 iundisj2 25066 disji2f 31808 disjpreima 31815 disjxpin 31819 iundisj2f 31821 disjunsn 31825 iundisj2fi 32008 disjxp1 43756 disjinfi 43891 |
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