Theorem disjlem14 39610

Description: Lemma for disjdmqseq 39617, partim2 39619 and petlem 39624 via disjlem17 39611, (general version of the former prtlem14 39708). (Contributed by Peter Mazsa, 10-Sep-2021.)

Assertion

Ref	Expression
disjlem14	⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))

Distinct variable group:   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)

Proof of Theorem disjlem14

Step	Hyp	Ref	Expression
1		dfdisjALTV5 39511	. . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ Rel 𝑅))
2	1	simplbi 502	. . 3 ⊢ ( Disj 𝑅 → ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
3		rsp2 3284	. . 3 ⊢ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)))
4	2, 3	syl 18	. 2 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)))
5		eceq1 8740	. . . 4 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
6	5	a1d 26	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
7		elin 3922	. . . 4 ⊢ (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) ↔ (𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅))
8		nel02 4292	. . . . 5 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ¬ 𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅))
9	8	pm2.21d 122	. . . 4 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
10	7, 9	biimtrrid 246	. . 3 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
11	6, 10	jaoi 871	. 2 ⊢ ((𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
12	4, 11	syl6 36	1 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ∩ cin 3905  ∅c0 4286  dom cdm 5663  Rel wrel 5668  [cec 8698   Disj wdisjALTV 38928

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8702  df-coss 39210  df-cnvrefrel 39316  df-disjALTV 39499

This theorem is used by:  disjlem17  39611