Theorem clwwlknondisjOLD 27292

Description: Obsolete version of clwwlknondisj 27287 as of 3-Mar-2022. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
clwwlknondisjOLD	⊢ Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}

Distinct variable groups:   𝑥,𝐺   𝑥,𝑁   𝑥,𝑉   𝑥,𝑤

Allowed substitution hints:   𝐺(𝑤)   𝑁(𝑤)   𝑉(𝑤)

Proof of Theorem clwwlknondisjOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inrab 4047	. . . . 5 ⊢ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)}
2		eqtr2 2791	. . . . . . . 8 ⊢ (((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦) → 𝑥 = 𝑦)
3	2	con3i 151	. . . . . . 7 ⊢ (¬ 𝑥 = 𝑦 → ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
4	3	ralrimivw 3116	. . . . . 6 ⊢ (¬ 𝑥 = 𝑦 → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
5		rabeq0 4103	. . . . . 6 ⊢ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅ ↔ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦))
6	4, 5	sylibr 224	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑥 ∧ (𝑤‘0) = 𝑦)} = ∅)
7	1, 6	syl5eq 2817	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
8	7	orri 849	. . 3 ⊢ (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
9	8	rgen2w 3074	. 2 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅)
10		eqeq2 2782	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑤‘0) = 𝑥 ↔ (𝑤‘0) = 𝑦))
11	10	rabbidv 3339	. . 3 ⊢ (𝑥 = 𝑦 → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦})
12	11	disjor 4768	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥} ∩ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑦}) = ∅))
13	9, 12	mpbir 221	1 ⊢ Disj 𝑥 ∈ 𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}

Syntax hints:  ¬ wn 3   ∧ wa 382   ∨ wo 834   = wceq 1631  ∀wral 3061  {crab 3065   ∩ cin 3722  ∅c0 4063  Disj wdisj 4754  ‘cfv 6031  (class class class)co 6793  0cc0 10138   ClWWalksN cclwwlkn 27174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rmo 3069  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-nul 4064  df-disj 4755

This theorem is referenced by: (None)