Theorem tr0elw 36782

Description: Every nonempty transitive set contains the empty set ∅ as an element, a consequence of Regularity. If we assume Transitive Containment, then we can omit the 𝐴 ∈ 𝑉 hypothesis, see tr0el 36783. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
tr0elw	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)

Proof of Theorem tr0elw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfreg 9530	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
2		trss 5207	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	imp 409	. . . . . 6 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
4		dfss 3914	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 ↔ 𝑥 = (𝑥 ∩ 𝐴))
5		eqeq2 2764	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐴) = ∅ → (𝑥 = (𝑥 ∩ 𝐴) ↔ 𝑥 = ∅))
6	4, 5	bitrid 285	. . . . . 6 ⊢ ((𝑥 ∩ 𝐴) = ∅ → (𝑥 ⊆ 𝐴 ↔ 𝑥 = ∅))
7	3, 6	syl5ibcom 247	. . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) = ∅ → 𝑥 = ∅))
8		eleq1 2840	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	8	biimpcd 251	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ∅ → ∅ ∈ 𝐴))
10	9	adantl 484	. . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 = ∅ → ∅ ∈ 𝐴))
11	7, 10	syld 47	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) = ∅ → ∅ ∈ 𝐴))
12	11	rexlimdva 3153	. . 3 ⊢ (Tr 𝐴 → (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ → ∅ ∈ 𝐴))
13	1, 12	syl5com 31	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (Tr 𝐴 → ∅ ∈ 𝐴))
14	13	3impia 1126	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076   ∩ cin 3894   ⊆ wss 3895  ∅c0 4276  Tr wtr 5197

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-ext 2724  ax-reg 9526

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-v 3446  df-dif 3898  df-in 3902  df-ss 3912  df-nul 4277  df-uni 4856  df-tr 5198

This theorem is referenced by:  ttc0elw  36825