Theorem elttcirr 36829

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 36828 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 3448	. . . . . 6 ⊢ 𝑦 ∈ V
2		inss2 4180	. . . . . . 7 ⊢ (𝐴 ∩ 𝑦) ⊆ 𝑦
3		ssn0 4348	. . . . . . 7 ⊢ (((𝐴 ∩ 𝑦) ⊆ 𝑦 ∧ (𝐴 ∩ 𝑦) ≠ ∅) → 𝑦 ≠ ∅)
4	2, 3	mpan 698	. . . . . 6 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → 𝑦 ≠ ∅)
5		zfreg 9530	. . . . . 6 ⊢ ((𝑦 ∈ V ∧ 𝑦 ≠ ∅) → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
6	1, 4, 5	sylancr 595	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅)
7		ineq1 4156	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 ∩ 𝑦) = (𝑥 ∩ 𝑦))
8	7	eqeq1d 2754	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑤 ∩ 𝑦) = ∅ ↔ (𝑥 ∩ 𝑦) = ∅))
9		ineq1 4156	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
10	9	eqeq1d 2754	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) = ∅ ↔ (𝐴 ∩ 𝑦) = ∅))
11	8, 10	rexraleqim 3597	. . . . 5 ⊢ ((∃𝑥 ∈ 𝑦 (𝑥 ∩ 𝑦) = ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
12	6, 11	sylan 588	. . . 4 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → (𝐴 ∩ 𝑦) = ∅)
13		neneq 2953	. . . . 5 ⊢ ((𝐴 ∩ 𝑦) ≠ ∅ → ¬ (𝐴 ∩ 𝑦) = ∅)
14	13	adantr 483	. . . 4 ⊢ (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)) → ¬ (𝐴 ∩ 𝑦) = ∅)
15	12, 14	pm2.65i 195	. . 3 ⊢ ¬ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))
16	15	nex 1810	. 2 ⊢ ¬ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))
17		eqeq2 2764	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑤 = 𝑥 ↔ 𝑤 = 𝐴))
18	17	imbi2d 342	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥) ↔ ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
19	18	ralbidv 3175	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥) ↔ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
20	19	anbi2d 638	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
21	20	exbidv 1931	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥)) ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
22		dfttc4 36828	. . . 4 ⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝑥))}
23	21, 22	elab2g 3630	. . 3 ⊢ (𝐴 ∈ TC+ 𝐴 → (𝐴 ∈ TC+ 𝐴 ↔ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴))))
24	23	ibi 269	. 2 ⊢ (𝐴 ∈ TC+ 𝐴 → ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑤 ∈ 𝑦 ((𝑤 ∩ 𝑦) = ∅ → 𝑤 = 𝐴)))
25	16, 24	mto 199	1 ⊢ ¬ 𝐴 ∈ TC+ 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132   ≠ wne 2947  ∀wral 3066  ∃wrex 3076  Vcvv 3444   ∩ cin 3894   ⊆ wss 3895  ∅c0 4276  TC+ cttc 36784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703  ax-reg 9526

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-ttc 36785

This theorem is referenced by: (None)