Theorem elttcirr 37241

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 37240 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 3454	. . . . . 6 ⊢ 𝑦 ∈ V
2		inss2 4182	. . . . . . 7 ⊢ (𝐴 ∩ 𝑦) ⊆ 𝑦
3		ssn0 4354	. . . . . . 7 ⊢ (((𝐴 ∩ 𝑦) ⊆ 𝑦 ∧ (𝐴 ∩ 𝑦) ≠ ∅) → 𝑦 ≠ ∅)
4	2, 3	mpan 703	. . . . . 6 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → 𝑦 ≠ ∅)
5		zfreg 9568	. . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
6	1, 4, 5	sylancr 599	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
7		ineq1 4158	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∩ 𝑦) = (𝑥 ∩ 𝑦))
8	7	eqeq1d 2762	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑤 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
9		ineq1 4158	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
10	9	eqeq1d 2762	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝐴 ∩ 𝑦) = ∅))
11	8, 10	rexraleqim 3600	. . . . 5 ⊢ ((∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
12	6, 11	sylan 592	. . . 4 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
13		neneq 2961	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ¬ (𝐴 ∩ 𝑦) = ∅)
14	13	adantr 486	. . . 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 1833	. 2 ⊢ ¬ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))
17		eqeq2 2772	. . . . . . . 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 3185	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥) ↔ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
20	19	anbi2d 642	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
21	20	exbidv 1954	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
22		dfttc4 37240	. . . 4 ⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥))}
23	21, 22	elab2g 3633	. . 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 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898  ∅c0 4278  TC+ cttc 37196

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734  ax-reg 9564

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-ttc 37197

This theorem is used by: (None)