Theorem 2onetap 7585

Description: Negated equality is a tight apartness on 2_o. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2onetap	⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o

Distinct variable group:   𝑣,𝑢

Proof of Theorem 2onetap

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2onn 6767	. . . . 5 ⊢ 2o ∈ ω
2		elnn 4733	. . . . 5 ⊢ ((𝑥 ∈ 2o ∧ 2o ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 425	. . . 4 ⊢ (𝑥 ∈ 2o → 𝑥 ∈ ω)
4		elnn 4733	. . . . 5 ⊢ ((𝑦 ∈ 2o ∧ 2o ∈ ω) → 𝑦 ∈ ω)
5	1, 4	mpan2 425	. . . 4 ⊢ (𝑦 ∈ 2o → 𝑦 ∈ ω)
6		nndceq 6745	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → DECID 𝑥 = 𝑦)
7	3, 5, 6	syl2an 289	. . 3 ⊢ ((𝑥 ∈ 2o ∧ 𝑦 ∈ 2o) → DECID 𝑥 = 𝑦)
8	7	rgen2 2630	. 2 ⊢ ∀𝑥 ∈ 2o ∀𝑦 ∈ 2o DECID 𝑥 = 𝑦
9		netap 7584	. 2 ⊢ (∀𝑥 ∈ 2o ∀𝑦 ∈ 2o DECID 𝑥 = 𝑦 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o)
10	8, 9	ax-mp 5	1 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o

Syntax hints:   ∧ wa 104  DECID wdc 842   ∈ wcel 2205   ≠ wne 2414  ∀wral 2522  {copab 4175  ωcom 4717  2_oc2o 6654   TAp wtap 7578

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-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-1o 6660  df-2o 6661  df-pap 7572  df-tap 7579

This theorem is referenced by:  2omotaplemst  7588