Theorem disjors 5094

Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjors	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))

Distinct variable groups:   𝑖,𝑗,𝑥,𝐴   𝐵,𝑖,𝑗

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjors

Step	Hyp	Ref	Expression
1		nfcv 2927	. . 3 ⊢ Ⅎ𝑖𝐵
2		nfcsb1v 3878	. . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
3		csbeq1a 3868	. . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
4	1, 2, 3	cbvdisj 5088	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵)
5		csbeq1 3857	. . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
6	5	disjor 5093	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
7	4, 6	bitri 278	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1570  ∀wral 3081  ⦋csb 3854   ∩ cin 3905  ∅c0 4286  Disj wdisj 5078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rmo 3371  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-in 3913  df-nul 4287  df-disj 5079

This theorem is used by:  disji2  5095  disjprg  5107  disjxiun  5108  disjxun  5109  iundisj2  25759  disji2f  32993  disjpreima  33000  disjxpin  33004  iundisj2f  33006  disjunsn  33010  iundisj2fi  33212  disjxp1  45847  disjinfi  45968