Theorem disjorsf 32925

Description: Two ways to say that a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Hypothesis

Ref	Expression
disjorsf.1	⊢ Ⅎ𝑥𝐴

Assertion

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

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjorsf

Step	Hyp	Ref	Expression
1		disjorsf.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2925	. . 3 ⊢ Ⅎ𝑖𝐵
3		nfcsb1v 3877	. . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
4		csbeq1a 3867	. . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
5	1, 2, 3, 4	cbvdisjf 32916	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵)
6		csbeq1 3856	. . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
7	6	disjor 5091	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
8	5, 7	bitri 278	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))

Syntax hints:   ↔ wb 209   ∨ wo 860   = wceq 1570  Ⅎwnfc 2910  ∀wral 3079  ⦋csb 3853   ∩ cin 3904  ∅c0 4286  Disj wdisj 5076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rmo 3369  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-in 3912  df-nul 4287  df-disj 5077

This theorem is referenced by:  disjif2  32926  disjdsct  33048