Theorem sn0disj 4127

Description: A collection, each of whose elements is empty or a singleton of the index, is disjoint. (Contributed by Jim Kingdon, 8-Oct-2026.)

Hypothesis

Ref	Expression
sn0disj.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ∅ ∨ 𝐵 = {𝑥}))

Assertion

Ref	Expression
sn0disj	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem sn0disj

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 533	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = ∅) → 𝑦 ∈ 𝐵)
2		simpr 110	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = ∅) → 𝐵 = ∅)
3	1, 2	eleqtrd 2317	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = ∅) → 𝑦 ∈ ∅)
4		noel 3525	. . . . . . . . . 10 ⊢ ¬ 𝑦 ∈ ∅
5	4	a1i 9	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = ∅) → ¬ 𝑦 ∈ ∅)
6	3, 5	pm2.21dd 629	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = ∅) → 𝑥 = 𝑦)
7	6	adantll 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝐵 = ∅) → 𝑥 = 𝑦)
8		simplr 533	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = {𝑥}) → 𝑦 ∈ 𝐵)
9		simpr 110	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑥})
10	8, 9	eleqtrd 2317	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = {𝑥}) → 𝑦 ∈ {𝑥})
11		velsn 3726	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
12	10, 11	sylib 122	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = {𝑥}) → 𝑦 = 𝑥)
13	12	equcomd 1759	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝐵 = {𝑥}) → 𝑥 = 𝑦)
14	13	adantll 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝐵 = {𝑥}) → 𝑥 = 𝑦)
15		sn0disj.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ∅ ∨ 𝐵 = {𝑥}))
16	15	adantrr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐵 = ∅ ∨ 𝐵 = {𝑥}))
17	7, 14, 16	mpjaodan 810	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 = 𝑦)
18	17	ex 115	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝑦))
19	18	alrimiv 1927	. . . 4 ⊢ (𝜑 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝑦))
20		mo2icl 3005	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 = 𝑦) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
21	19, 20	syl 14	. . 3 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
22	21	alrimiv 1927	. 2 ⊢ (𝜑 → ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
23		dfdisj2 4108	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
24	22, 23	sylibr 134	1 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  ∀wal 1400   = wceq 1402  ∃*wmo 2087   ∈ wcel 2209  ∅c0 3520  {csn 3709  Disj wdisj 4106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rmo 2536  df-v 2823  df-dif 3222  df-nul 3521  df-sn 3715  df-disj 4107

This theorem is used by:  gsum2d2fi  14274