Theorem disjnims 4099

Description: If a collection 𝐵(𝑖) for 𝑖 ∈ 𝐴 is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.)

Assertion

Ref	Expression
disjnims	⊢ (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 disjnims

Step	Hyp	Ref	Expression
1		nfcv 2384	. . 3 ⊢ Ⅎ𝑖𝐵
2		nfcsb1v 3170	. . 3 ⊢ Ⅎ𝑥⦋𝑖 / 𝑥⦌𝐵
3		csbeq1a 3146	. . 3 ⊢ (𝑥 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑥⦌𝐵)
4	1, 2, 3	cbvdisj 4094	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵)
5		csbeq1 3140	. . 3 ⊢ (𝑖 = 𝑗 → ⦋𝑖 / 𝑥⦌𝐵 = ⦋𝑗 / 𝑥⦌𝐵)
6	5	disjnim 4098	. 2 ⊢ (Disj 𝑖 ∈ 𝐴 ⦋𝑖 / 𝑥⦌𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))
7	4, 6	sylbi 121	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 ≠ 𝑗 → (⦋𝑖 / 𝑥⦌𝐵 ∩ ⦋𝑗 / 𝑥⦌𝐵) = ∅))

Syntax hints:   → wi 4   = wceq 1398   ≠ wne 2412  ∀wral 2520  ⦋csb 3137   ∩ cin 3209  ∅c0 3507  Disj wdisj 4084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-in 3216  df-nul 3508  df-disj 4085

This theorem is referenced by:  disji2  4100