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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjorsf | Structured version Visualization version GIF version |
Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
Ref | Expression |
---|---|
disjorsf.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
disjorsf | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjorsf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2976 | . . 3 ⊢ Ⅎ𝑖𝐵 | |
3 | nfcsb1v 3900 | . . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵 | |
4 | csbeq1a 3890 | . . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵) | |
5 | 1, 2, 3, 4 | cbvdisjf 30321 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵) |
6 | csbeq1 3879 | . . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵) | |
7 | 6 | disjor 5039 | . 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅)) |
8 | 5, 7 | bitri 277 | 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 208 ∨ wo 843 = wceq 1536 Ⅎwnfc 2960 ∀wral 3137 ⦋csb 3876 ∩ cin 3928 ∅c0 4284 Disj wdisj 5024 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rmo 3145 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-in 3936 df-nul 4285 df-disj 5025 |
This theorem is referenced by: disjif2 30331 disjdsct 30438 |
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