Theorem 2onetap 7437

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 6665	. . . . 5 ⊢ 2o ∈ ω
2		elnn 4697	. . . . 5 ⊢ ((𝑥 ∈ 2o ∧ 2o ∈ ω) → 𝑥 ∈ ω)
3	1, 2	mpan2 425	. . . 4 ⊢ (𝑥 ∈ 2o → 𝑥 ∈ ω)
4		elnn 4697	. . . . 5 ⊢ ((𝑦 ∈ 2o ∧ 2o ∈ ω) → 𝑦 ∈ ω)
5	1, 4	mpan2 425	. . . 4 ⊢ (𝑦 ∈ 2o → 𝑦 ∈ ω)
6		nndceq 6643	. . . 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 2616	. 2 ⊢ ∀𝑥 ∈ 2o ∀𝑦 ∈ 2o DECID 𝑥 = 𝑦
9		netap 7436	. 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 839   ∈ wcel 2200   ≠ wne 2400  ∀wral 2508  {copab 4143  ωcom 4681  2_oc2o 6554   TAp wtap 7431

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-1o 6560  df-2o 6561  df-pap 7430  df-tap 7432

This theorem is referenced by:  2omotaplemst  7440