Theorem 2omotaplemst 7520

Description: Lemma for 2omotap 7521. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotaplemst	⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)

Distinct variable group:   𝜑,𝑟

Proof of Theorem 2omotaplemst

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2oneel 7518	. . . 4 ⊢ ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)}
2		2omotaplemap 7519	. . . . . 6 ⊢ (¬ ¬ 𝜑 → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
3	2	adantl 277	. . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o)
4		2onn 6732	. . . . . . . . . 10 ⊢ 2o ∈ ω
5	4	elexi 2816	. . . . . . . . 9 ⊢ 2o ∈ V
6	5, 5	xpex 4848	. . . . . . . 8 ⊢ (2o × 2o) ∈ V
7		opabssxp 4806	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ⊆ (2o × 2o)
8	6, 7	ssexi 4232	. . . . . . 7 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ∈ V
9	8	a1i 9	. . . . . 6 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ∈ V)
10		opabssxp 4806	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ⊆ (2o × 2o)
11	6, 10	ssexi 4232	. . . . . . 7 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ∈ V
12	11	a1i 9	. . . . . 6 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ∈ V)
13		simpl 109	. . . . . 6 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → ∃*𝑟 𝑟 TAp 2o)
14		2onetap 7517	. . . . . . 7 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o
15	14	a1i 9	. . . . . 6 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o)
16		tapeq1 7514	. . . . . . 7 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} → (𝑟 TAp 2o ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o))
17		tapeq1 7514	. . . . . . 7 ⊢ (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} → (𝑟 TAp 2o ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o))
18	16, 17	mob 2989	. . . . . 6 ⊢ ((({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ∈ V ∧ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ∈ V) ∧ ∃*𝑟 𝑟 TAp 2o ∧ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} TAp 2o) → ({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o))
19	9, 12, 13, 15, 18	syl211anc 1280	. . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → ({⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ↔ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} TAp 2o))
20	3, 19	mpbird 167	. . . 4 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))})
21	1, 20	eleqtrid 2320	. . 3 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))})
22		0lt2o 6652	. . . 4 ⊢ ∅ ∈ 2o
23		1lt2o 6653	. . . 4 ⊢ 1o ∈ 2o
24		neeq1 2416	. . . . . 6 ⊢ (𝑢 = ∅ → (𝑢 ≠ 𝑣 ↔ ∅ ≠ 𝑣))
25	24	anbi2d 464	. . . . 5 ⊢ (𝑢 = ∅ → ((𝜑 ∧ 𝑢 ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 𝑣)))
26		neeq2 2417	. . . . . 6 ⊢ (𝑣 = 1o → (∅ ≠ 𝑣 ↔ ∅ ≠ 1o))
27	26	anbi2d 464	. . . . 5 ⊢ (𝑣 = 1o → ((𝜑 ∧ ∅ ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 1o)))
28	25, 27	opelopab2 4371	. . . 4 ⊢ ((∅ ∈ 2o ∧ 1o ∈ 2o) → (⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ↔ (𝜑 ∧ ∅ ≠ 1o)))
29	22, 23, 28	mp2an 426	. . 3 ⊢ (⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ↔ (𝜑 ∧ ∅ ≠ 1o))
30	21, 29	sylib 122	. 2 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → (𝜑 ∧ ∅ ≠ 1o))
31	30	simpld 112	1 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃*wmo 2080   ∈ wcel 2202   ≠ wne 2403  Vcvv 2803  ∅c0 3496  ⟨cop 3676  {copab 4154  ωcom 4694   × cxp 4729  1_oc1o 6618  2_oc2o 6619   TAp wtap 7511

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-1o 6625  df-2o 6626  df-pap 7510  df-tap 7512

This theorem is referenced by:  2omotap  7521