Theorem disjors 5129

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 2902	. . 3 ⊢ Ⅎ𝑖𝐵
2		nfcsb1v 3918	. . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
3		csbeq1a 3907	. . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
4	1, 2, 3	cbvdisj 5123	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵)
5		csbeq1 3896	. . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
6	5	disjor 5128	. 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) = ∅))

Syntax hints:   ↔ wb 205   ∨ wo 844   = wceq 1540  ∀wral 3060  ⦋csb 3893   ∩ cin 3947  ∅c0 4322  Disj wdisj 5113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rmo 3375  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-in 3955  df-nul 4323  df-disj 5114

This theorem is referenced by:  disji2  5130  disjprgw  5143  disjprg  5144  disjxiun  5145  disjxun  5146  iundisj2  25399  disji2f  32243  disjpreima  32250  disjxpin  32254  iundisj2f  32256  disjunsn  32260  iundisj2fi  32443  disjxp1  44221  disjinfi  44353