Theorem 2omotaplemst 7476

Description: Lemma for 2omotap 7477. (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 7474	. . . 4 ⊢ ⟨∅, 1o⟩ ∈ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)}
2		2omotaplemap 7475	. . . . . 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 6688	. . . . . . . . . 10 ⊢ 2o ∈ ω
5	4	elexi 2815	. . . . . . . . 9 ⊢ 2o ∈ V
6	5, 5	xpex 4842	. . . . . . . 8 ⊢ (2o × 2o) ∈ V
7		opabssxp 4800	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ 𝑢 ≠ 𝑣)} ⊆ (2o × 2o)
8	6, 7	ssexi 4227	. . . . . . 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 4800	. . . . . . . 8 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 2o ∧ 𝑣 ∈ 2o) ∧ (𝜑 ∧ 𝑢 ≠ 𝑣))} ⊆ (2o × 2o)
11	6, 10	ssexi 4227	. . . . . . 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 7473	. . . . . . 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 7470	. . . . . . 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 7470	. . . . . . 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 2988	. . . . . 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 1279	. . . . 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 6608	. . . 4 ⊢ ∅ ∈ 2o
23		1lt2o 6609	. . . 4 ⊢ 1o ∈ 2o
24		neeq1 2415	. . . . . 6 ⊢ (𝑢 = ∅ → (𝑢 ≠ 𝑣 ↔ ∅ ≠ 𝑣))
25	24	anbi2d 464	. . . . 5 ⊢ (𝑢 = ∅ → ((𝜑 ∧ 𝑢 ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 𝑣)))
26		neeq2 2416	. . . . . 6 ⊢ (𝑣 = 1o → (∅ ≠ 𝑣 ↔ ∅ ≠ 1o))
27	26	anbi2d 464	. . . . 5 ⊢ (𝑣 = 1o → ((𝜑 ∧ ∅ ≠ 𝑣) ↔ (𝜑 ∧ ∅ ≠ 1o)))
28	25, 27	opelopab2 4365	. . . 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 1397  ∃*wmo 2080   ∈ wcel 2202   ≠ wne 2402  Vcvv 2802  ∅c0 3494  ⟨cop 3672  {copab 4149  ωcom 4688   × cxp 4723  1_oc1o 6574  2_oc2o 6575   TAp wtap 7467

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-1o 6581  df-2o 6582  df-pap 7466  df-tap 7468

This theorem is referenced by:  2omotap  7477