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| 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 2905 | . . 3 ⊢ Ⅎ𝑖𝐵 | |
| 2 | nfcsb1v 3923 | . . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
| 3 | csbeq1a 3913 | . . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
| 4 | 1, 2, 3 | cbvdisj 5120 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵) | 
| 5 | csbeq1 3902 | . . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
| 6 | 5 | disjor 5125 | . 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 206 ∨ wo 848 = wceq 1540 ∀wral 3061 ⦋csb 3899 ∩ cin 3950 ∅c0 4333 Disj wdisj 5110 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rmo 3380 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-in 3958 df-nul 4334 df-disj 5111 | 
| This theorem is referenced by: disji2 5127 disjprg 5139 disjxiun 5140 disjxun 5141 iundisj2 25584 disji2f 32590 disjpreima 32597 disjxpin 32601 iundisj2f 32603 disjunsn 32607 iundisj2fi 32799 disjxp1 45074 disjinfi 45197 | 
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