Theorem 2onetap 7569

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 6754	. . . . 5 ⊢ 2o ∈ ω
2		elnn 4728	. . . . 5 ⊢ ((𝑥 ∈ 2o ∧ 2o ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 425	. . . 4 ⊢ (𝑥 ∈ 2o → 𝑥 ∈ ω)
4		elnn 4728	. . . . 5 ⊢ ((𝑦 ∈ 2o ∧ 2o ∈ ω) → 𝑦 ∈ ω)
5	1, 4	mpan2 425	. . . 4 ⊢ (𝑦 ∈ 2o → 𝑦 ∈ ω)
6		nndceq 6732	. . . 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 2628	. 2 ⊢ ∀𝑥 ∈ 2o ∀𝑦 ∈ 2o DECID 𝑥 = 𝑦
9		netap 7568	. 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 2203   ≠ wne 2412  ∀wral 2520  {copab 4170  ωcom 4712  2_oc2o 6641   TAp wtap 7563

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-1o 6647  df-2o 6648  df-pap 7559  df-tap 7564

This theorem is referenced by:  2omotaplemst  7572