Theorem exmidmotap 7523

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 531	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 TAp 𝑥)
2		exmidapne 7522	. . . . . . . . 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 533	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑠 TAp 𝑥)
6		exmidapne 7522	. . . . . . . . 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 2267	. . . . . 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 1923	. . . 4 ⊢ (EXMID → ∀𝑟∀𝑠((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠))
12		tapeq1 7514	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 TAp 𝑥 ↔ 𝑠 TAp 𝑥))
13	12	mo4 2141	. . . 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 1922	. 2 ⊢ (EXMID → ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
16		2onn 6732	. . . 4 ⊢ 2o ∈ ω
17		tapeq2 7515	. . . . . 6 ⊢ (𝑥 = 2o → (𝑟 TAp 𝑥 ↔ 𝑟 TAp 2o))
18	17	mobidv 2115	. . . . 5 ⊢ (𝑥 = 2o → (∃*𝑟 𝑟 TAp 𝑥 ↔ ∃*𝑟 𝑟 TAp 2o))
19	18	spcgv 2894	. . . 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 7521	. . 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 1396   = wceq 1398  ∃*wmo 2080   ∈ wcel 2202   ≠ wne 2403  {copab 4154  EXMIDwem 4290  ωcom 4694  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-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  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-rab 2520  df-v 2805  df-sbc 3033  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-mpt 4157  df-tr 4193  df-exmid 4291  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-pap 7510  df-tap 7512

This theorem is referenced by: (None)