Theorem 2omotaplemst 7325

Description: Lemma for 2omotap 7326. (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 7323	. . . 4 ⊢ ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)}
2		2omotaplemap 7324	. . . . . 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 6579	. . . . . . . . . 10 ⊢ 2o ∈ ω
5	4	elexi 2775	. . . . . . . . 9 ⊢ 2o ∈ V
6	5, 5	xpex 4778	. . . . . . . 8 ⊢ (2o × 2o) ∈ V
7		opabssxp 4737	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ⊆ (2o × 2o)
8	6, 7	ssexi 4171	. . . . . . 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 4737	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ⊆ (2o × 2o)
11	6, 10	ssexi 4171	. . . . . . 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 7322	. . . . . . 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 7319	. . . . . . 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 7319	. . . . . . 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 2946	. . . . . 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 1255	. . . . 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 2285	. . 3 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝜑) → ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))})
22		0lt2o 6499	. . . 4 ⊢ ∅ ∈ 2o
23		1lt2o 6500	. . . 4 ⊢ 1o ∈ 2o
24		neeq1 2380	. . . . . 6 ⊢ (𝑢 = ∅ → (𝑢 ≠ 𝑣 ↔ ∅ ≠ 𝑣))
25	24	anbi2d 464	. . . . 5 ⊢ (𝑢 = ∅ → ((𝜑 ∧ 𝑢 ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 𝑣)))
26		neeq2 2381	. . . . . 6 ⊢ (𝑣 = 1o → (∅ ≠ 𝑣 ↔ ∅ ≠ 1o))
27	26	anbi2d 464	. . . . 5 ⊢ (𝑣 = 1o → ((𝜑 ∧ ∅ ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 1o)))
28	25, 27	opelopab2 4305	. . . 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 1364  ∃*wmo 2046   ∈ wcel 2167   ≠ wne 2367  Vcvv 2763  ∅c0 3450  ⟨cop 3625  {copab 4093  ωcom 4626   × cxp 4661  1_oc1o 6467  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:  2omotap  7326