Theorem exmidmotap 7617

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 535	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑟 TAp 𝑥)
2		exmidapne 7616	. . . . . . . . 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 537	. . . . . . . 8 ⊢ ((EXMID ∧ (𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥)) → 𝑠 TAp 𝑥)
6		exmidapne 7616	. . . . . . . . 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 2274	. . . . . 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 1928	. . . 4 ⊢ (EXMID → ∀𝑟∀𝑠((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠))
12		tapeq1 7608	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 TAp 𝑥 ↔ 𝑠 TAp 𝑥))
13	12	mo4 2148	. . . 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 1927	. 2 ⊢ (EXMID → ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
16		2onn 6784	. . . 4 ⊢ 2o ∈ ω
17		tapeq2 7609	. . . . . 6 ⊢ (𝑥 = 2o → (𝑟 TAp 𝑥 ↔ 𝑟 TAp 2o))
18	17	mobidv 2122	. . . . 5 ⊢ (𝑥 = 2o → (∃*𝑟 𝑟 TAp 𝑥 ↔ ∃*𝑟 𝑟 TAp 2o))
19	18	spcgv 2912	. . . 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 7615	. . 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 1400   = wceq 1402  ∃*wmo 2087   ∈ wcel 2209   ≠ wne 2420  {copab 4186  EXMIDwem 4326  ωcom 4732  2_oc2o 6671   TAp wtap 7604

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-exmid 4327  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-pap 7598  df-tap 7605

This theorem is referenced by: (None)