Theorem elttcirr 37070

Description: Irreflexivity of 𝐴 ∈ TC+ 𝐵 relationship. This is a consequence of Regularity, but it does not require Transitive Containment. We use the alternative expression dfttc4 37069 to construct a set in which 𝐴 is both ∈-minimal and not ∈-minimal. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
elttcirr	⊢ ¬ 𝐴 ∈ TC+ 𝐴

Proof of Theorem elttcirr

Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
2		inss2 4189	. . . . . . 7 ⊢ (𝐴 ∩ 𝑦) ⊆ 𝑦
3		ssn0 4361	. . . . . . 7 ⊢ (((𝐴 ∩ 𝑦) ⊆ 𝑦 ∧ (𝐴 ∩ 𝑦) ≠ ∅) → 𝑦 ≠ ∅)
4	2, 3	mpan 702	. . . . . 6 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → 𝑦 ≠ ∅)
5		zfreg 9556	. . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
6	1, 4, 5	sylancr 598	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
7		ineq1 4165	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∩ 𝑦) = (𝑥 ∩ 𝑦))
8	7	eqeq1d 2764	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑤 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
9		ineq1 4165	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
10	9	eqeq1d 2764	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝐴 ∩ 𝑦) = ∅))
11	8, 10	rexraleqim 3605	. . . . 5 ⊢ ((∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
12	6, 11	sylan 591	. . . 4 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
13		neneq 2963	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ¬ (𝐴 ∩ 𝑦) = ∅)
14	13	adantr 485	. . . 4 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → ¬ (𝐴 ∩ 𝑦) = ∅)
15	12, 14	pm2.65i 196	. . 3 ⊢ ¬ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))
16	15	nex 1829	. 2 ⊢ ¬ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))
17		eqeq2 2774	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑤 = 𝑥 ↔ 𝑤 = 𝐴))
18	17	imbi2d 343	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥) ↔ ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
19	18	ralbidv 3187	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥) ↔ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
20	19	anbi2d 641	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
21	20	exbidv 1950	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
22		dfttc4 37069	. . . 4 ⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥))}
23	21, 22	elab2g 3638	. . 3 ⊢ (𝐴 ∈ TC+ 𝐴 → (𝐴 ∈ TC+ 𝐴 ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
24	23	ibi 270	. 2 ⊢ (𝐴 ∈ TC+ 𝐴 → ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
25	16, 24	mto 200	1 ⊢ ¬ 𝐴 ∈ TC+ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  TC+ cttc 37025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-ttc 37026

This theorem is used by: (None)