Theorem exmidmotap 7344

Description: The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)

Assertion

Ref	Expression
exmidmotap	⊢ (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥)

Distinct variable group:   𝑥,𝑟

Proof of Theorem exmidmotap

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

Step	Hyp	Ref	Expression
1		simprl 529	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 TAp 𝑥)
2		exmidapne 7343	. . . . . . . . 9 ⊢ (EXMID → (𝑟 TAp 𝑥 ↔ 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)}))
3	2	adantr 276	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → (𝑟 TAp 𝑥 ↔ 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)}))
4	1, 3	mpbid 147	. . . . . . 7 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)})
5		simprr 531	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑠 TAp 𝑥)
6		exmidapne 7343	. . . . . . . . 9 ⊢ (EXMID → (𝑠 TAp 𝑥 ↔ 𝑠 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)}))
7	6	adantr 276	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → (𝑠 TAp 𝑥 ↔ 𝑠 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)}))
8	5, 7	mpbid 147	. . . . . . 7 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑠 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ∧ 𝑢 ≠ 𝑣)})
9	4, 8	eqtr4d 2232	. . . . . 6 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 = 𝑠)
10	9	ex 115	. . . . 5 ⊢ (EXMID → ((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠))
11	10	alrimivv 1889	. . . 4 ⊢ (EXMID → ∀𝑟∀𝑠((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠))
12		tapeq1 7335	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 TAp 𝑥 ↔ 𝑠 TAp 𝑥))
13	12	mo4 2106	. . . 4 ⊢ (∃*𝑟 𝑟 TAp 𝑥 ↔ ∀𝑟∀𝑠((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠))
14	11, 13	sylibr 134	. . 3 ⊢ (EXMID → ∃*𝑟 𝑟 TAp 𝑥)
15	14	alrimiv 1888	. 2 ⊢ (EXMID → ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
16		2onn 6588	. . . 4 ⊢ 2o ∈ ω
17		tapeq2 7336	. . . . . 6 ⊢ (𝑥 = 2o → (𝑟 TAp 𝑥 ↔ 𝑟 TAp 2o))
18	17	mobidv 2081	. . . . 5 ⊢ (𝑥 = 2o → (∃*𝑟 𝑟 TAp 𝑥 ↔ ∃*𝑟 𝑟 TAp 2o))
19	18	spcgv 2851	. . . 4 ⊢ (2o ∈ ω → (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → ∃*𝑟 𝑟 TAp 2o))
20	16, 19	ax-mp 5	. . 3 ⊢ (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → ∃*𝑟 𝑟 TAp 2o)
21		2omotap 7342	. . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)
22	20, 21	syl 14	. 2 ⊢ (∀𝑥∃*𝑟 𝑟 TAp 𝑥 → EXMID)
23	15, 22	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥∃*𝑟 𝑟 TAp 𝑥)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃*wmo 2046   ∈ wcel 2167   ≠ wne 2367  {copab 4094  EXMIDwem 4228  ωcom 4627  2_oc2o 6477   TAp wtap 7332

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 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  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-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-exmid 4229  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-2o 6484  df-pap 7331  df-tap 7333

This theorem is referenced by: (None)