Theorem tr0elw 37090

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 37091. (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 9571	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
2		trss 5226	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	imp 412	. . . . . 6 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
4		dfss 3921	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 ↔ 𝑥 = (𝑥 ∩ 𝐴))
5		eqeq2 2774	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐴) = ∅ → (𝑥 = (𝑥 ∩ 𝐴) ↔ 𝑥 = ∅))
6	4, 5	bitrid 286	. . . . . 6 ⊢ ((𝑥 ∩ 𝐴) = ∅ → (𝑥 ⊆ 𝐴 ↔ 𝑥 = ∅))
7	3, 6	syl5ibcom 248	. . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) = ∅ → 𝑥 = ∅))
8		eleq1 2850	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	8	biimpcd 252	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ∅ → ∅ ∈ 𝐴))
10	9	adantl 487	. . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 = ∅ → ∅ ∈ 𝐴))
11	7, 10	syld 48	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) = ∅ → ∅ ∈ 𝐴))
12	11	rexlimdva 3165	. . 3 ⊢ (Tr 𝐴 → (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ → ∅ ∈ 𝐴))
13	1, 12	syl5com 32	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (Tr 𝐴 → ∅ ∈ 𝐴))
14	13	3impia 1135	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  Tr wtr 5216

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-ext 2734  ax-reg 9567

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4283  df-uni 4871  df-tr 5217

This theorem is used by:  ttc0elw  37133