Theorem 2onetap 7627

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 6798	. . . . 5 ⊢ 2o ∈ ω
2		elnn 4755	. . . . 5 ⊢ ((𝑥 ∈ 2o ∧ 2o ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 429	. . . 4 ⊢ (𝑥 ∈ 2o → 𝑥 ∈ ω)
4		elnn 4755	. . . . 5 ⊢ ((𝑦 ∈ 2o ∧ 2o ∈ ω) → 𝑦 ∈ ω)
5	1, 4	mpan2 429	. . . 4 ⊢ (𝑦 ∈ 2o → 𝑦 ∈ ω)
6		nndceq 6776	. . . 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 2636	. 2 ⊢ ∀𝑥 ∈ 2o ∀𝑦 ∈ 2o DECID 𝑥 = 𝑦
9		netap 7626	. 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

This proof depends on syntax axioms:   ∧ wa 104  DECID wdc 846   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  {copab 4192  ωcom 4739  2_oc2o 6685   TAp wtap 7620

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-tr 4232  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-1o 6691  df-2o 6692  df-pap 7614  df-tap 7621

This theorem is used by:  2omotaplemst  7630