Theorem elttcirr 36719

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 36718 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 3434	. . . . . 6 ⊢ 𝑦 ∈ V
2		inss2 4179	. . . . . . 7 ⊢ (𝐴 ∩ 𝑦) ⊆ 𝑦
3		ssn0 4345	. . . . . . 7 ⊢ (((𝐴 ∩ 𝑦) ⊆ 𝑦 ∧ (𝐴 ∩ 𝑦) ≠ ∅) → 𝑦 ≠ ∅)
4	2, 3	mpan 691	. . . . . 6 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → 𝑦 ≠ ∅)
5		zfreg 9502	. . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
6	1, 4, 5	sylancr 588	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
7		ineq1 4154	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∩ 𝑦) = (𝑥 ∩ 𝑦))
8	7	eqeq1d 2739	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑤 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
9		ineq1 4154	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
10	9	eqeq1d 2739	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝐴 ∩ 𝑦) = ∅))
11	8, 10	rexraleqim 3590	. . . . 5 ⊢ ((∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
12	6, 11	sylan 581	. . . 4 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
13		neneq 2939	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ¬ (𝐴 ∩ 𝑦) = ∅)
14	13	adantr 480	. . . 4 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → ¬ (𝐴 ∩ 𝑦) = ∅)
15	12, 14	pm2.65i 194	. . 3 ⊢ ¬ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))
16	15	nex 1802	. 2 ⊢ ¬ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))
17		eqeq2 2749	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑤 = 𝑥 ↔ 𝑤 = 𝐴))
18	17	imbi2d 340	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥) ↔ ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
19	18	ralbidv 3161	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥) ↔ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
20	19	anbi2d 631	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
21	20	exbidv 1923	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
22		dfttc4 36718	. . . 4 ⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥))}
23	21, 22	elab2g 3624	. . 3 ⊢ (𝐴 ∈ TC+ 𝐴 → (𝐴 ∈ TC+ 𝐴 ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
24	23	ibi 267	. 2 ⊢ (𝐴 ∈ TC+ 𝐴 → ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
25	16, 24	mto 197	1 ⊢ ¬ 𝐴 ∈ TC+ 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  TC+ cttc 36674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680  ax-reg 9498

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-ttc 36675

This theorem is referenced by: (None)