Theorem 2onetap 7322

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 6579	. . . . 5 ⊢ 2o ∈ ω
2		elnn 4642	. . . . 5 ⊢ ((𝑥 ∈ 2o ∧ 2o ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 425	. . . 4 ⊢ (𝑥 ∈ 2o → 𝑥 ∈ ω)
4		elnn 4642	. . . . 5 ⊢ ((𝑦 ∈ 2o ∧ 2o ∈ ω) → 𝑦 ∈ ω)
5	1, 4	mpan2 425	. . . 4 ⊢ (𝑦 ∈ 2o → 𝑦 ∈ ω)
6		nndceq 6557	. . . 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 2583	. 2 ⊢ ∀𝑥 ∈ 2o ∀𝑦 ∈ 2o DECID 𝑥 = 𝑦
9		netap 7321	. 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 835   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  {copab 4093  ωcom 4626  2_oc2o 6468   TAp wtap 7316

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-1o 6474  df-2o 6475  df-pap 7315  df-tap 7317

This theorem is referenced by:  2omotaplemst  7325