Theorem disjlem14 39400

Description: Lemma for disjdmqseq 39407, partim2 39409 and petlem 39414 via disjlem17 39401, (general version of the former prtlem14 39498). (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 39301	. . . 4 ⊢ ( Disj 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ Rel 𝑅))
2	1	simplbi 500	. . 3 ⊢ ( Disj 𝑅 → ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
3		rsp2 3279	. . 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 17	. 2 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → (𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)))
5		eceq1 8718	. . . 4 ⊢ (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
6	5	a1d 25	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
7		elin 3920	. . . 4 ⊢ (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) ↔ (𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅))
8		nel02 4291	. . . . 5 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ¬ 𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅))
9	8	pm2.21d 121	. . . 4 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
10	7, 9	biimtrrid 245	. . 3 ⊢ (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
11	6, 10	jaoi 868	. 2 ⊢ ((𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
12	4, 11	syl6 35	1 ⊢ ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅 ∧ 𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅 ∧ 𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∩ cin 3903  ∅c0 4285  dom cdm 5647  Rel wrel 5652  [cec 8676   Disj wdisjALTV 38718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rmo 3367  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8680  df-coss 39000  df-cnvrefrel 39106  df-disjALTV 39289

This theorem is referenced by:  disjlem17  39401