Theorem disjorsf 29393

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 2764	. . 3 ⊢ Ⅎ𝑖𝐵
3		nfcsb1v 3549	. . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
4		csbeq1a 3542	. . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
5	1, 2, 3, 4	cbvdisjf 29385	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵)
6		csbeq1 3536	. . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
7	6	disjor 4634	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
8	5, 7	bitri 264	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))

Syntax hints:   ↔ wb 196   ∨ wo 383   = wceq 1483  Ⅎwnfc 2751  ∀wral 2912  ⦋csb 3533   ∩ cin 3573  ∅c0 3915  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  disjif2  29394  disjdsct  29480