Theorem 2omotaplemst 7257

Description: Lemma for 2omotap 7258. (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 7255	. . . 4 ⊢ ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)}
2		2omotaplemap 7256	. . . . . 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 6522	. . . . . . . . . 10 ⊢ 2o ∈ ω
5	4	elexi 2750	. . . . . . . . 9 ⊢ 2o ∈ V
6	5, 5	xpex 4742	. . . . . . . 8 ⊢ (2o × 2o) ∈ V
7		opabssxp 4701	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ⊆ (2o × 2o)
8	6, 7	ssexi 4142	. . . . . . 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 4701	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ⊆ (2o × 2o)
11	6, 10	ssexi 4142	. . . . . . 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 7254	. . . . . . 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 7251	. . . . . . 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 7251	. . . . . . 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 2920	. . . . . 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 1244	. . . . 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 2266	. . 3 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))})
22		0lt2o 6442	. . . 4 ⊢ ∅ ∈ 2o
23		1lt2o 6443	. . . 4 ⊢ 1o ∈ 2o
24		neeq1 2360	. . . . . 6 ⊢ (𝑢 = ∅ → (𝑢 ≠ 𝑣 ↔ ∅ ≠ 𝑣))
25	24	anbi2d 464	. . . . 5 ⊢ (𝑢 = ∅ → ((𝜑 ∧ 𝑢 ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 𝑣)))
26		neeq2 2361	. . . . . 6 ⊢ (𝑣 = 1o → (∅ ≠ 𝑣 ↔ ∅ ≠ 1o))
27	26	anbi2d 464	. . . . 5 ⊢ (𝑣 = 1o → ((𝜑 ∧ ∅ ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 1o)))
28	25, 27	opelopab2 4271	. . . 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 1353  ∃*wmo 2027   ∈ wcel 2148   ≠ wne 2347  Vcvv 2738  ∅c0 3423  ⟨cop 3596  {copab 4064  ωcom 4590   × cxp 4625  1_oc1o 6410  2_oc2o 6411   TAp wtap 7248

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-1o 6417  df-2o 6418  df-pap 7247  df-tap 7249

This theorem is referenced by:  2omotap  7258