Theorem exmidmotap 7403

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 7402	. . . . . . . . 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 7402	. . . . . . . . 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 2242	. . . . . 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 1899	. . . 4 ⊢ (EXMID → ∀𝑟∀𝑠((𝑟 TAp 𝑥 ∧ 𝑠 TAp 𝑥) → 𝑟 = 𝑠))
12		tapeq1 7394	. . . . 5 ⊢ (𝑟 = 𝑠 → (𝑟 TAp 𝑥 ↔ 𝑠 TAp 𝑥))
13	12	mo4 2116	. . . 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 1898	. 2 ⊢ (EXMID → ∀𝑥∃*𝑟 𝑟 TAp 𝑥)
16		2onn 6625	. . . 4 ⊢ 2o ∈ ω
17		tapeq2 7395	. . . . . 6 ⊢ (𝑥 = 2o → (𝑟 TAp 𝑥 ↔ 𝑟 TAp 2o))
18	17	mobidv 2091	. . . . 5 ⊢ (𝑥 = 2o → (∃*𝑟 𝑟 TAp 𝑥 ↔ ∃*𝑟 𝑟 TAp 2o))
19	18	spcgv 2864	. . . 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 7401	. . 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 1371   = wceq 1373  ∃*wmo 2056   ∈ wcel 2177   ≠ wne 2377  {copab 4115  EXMIDwem 4249  ωcom 4651  2_oc2o 6514   TAp wtap 7391

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-exmid 4250  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-fo 5291  df-fv 5293  df-1st 6244  df-2nd 6245  df-1o 6520  df-2o 6521  df-pap 7390  df-tap 7392

This theorem is referenced by: (None)