Theorem tr0elw 36672

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 36673. (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 9502	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
2		trss 5203	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3	2	imp 406	. . . . . 6 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
4		dfss 3909	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 ↔ 𝑥 = (𝑥 ∩ 𝐴))
5		eqeq2 2749	. . . . . . 7 ⊢ ((𝑥 ∩ 𝐴) = ∅ → (𝑥 = (𝑥 ∩ 𝐴) ↔ 𝑥 = ∅))
6	4, 5	bitrid 283	. . . . . 6 ⊢ ((𝑥 ∩ 𝐴) = ∅ → (𝑥 ⊆ 𝐴 ↔ 𝑥 = ∅))
7	3, 6	syl5ibcom 245	. . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) = ∅ → 𝑥 = ∅))
8		eleq1 2825	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝐴 ↔ ∅ ∈ 𝐴))
9	8	biimpcd 249	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ∅ → ∅ ∈ 𝐴))
10	9	adantl 481	. . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 = ∅ → ∅ ∈ 𝐴))
11	7, 10	syld 47	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∩ 𝐴) = ∅ → ∅ ∈ 𝐴))
12	11	rexlimdva 3139	. . 3 ⊢ (Tr 𝐴 → (∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅ → ∅ ∈ 𝐴))
13	1, 12	syl5com 31	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (Tr 𝐴 → ∅ ∈ 𝐴))
14	13	3impia 1118	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  Tr wtr 5193

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-ext 2709  ax-reg 9498

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-v 3432  df-dif 3893  df-in 3897  df-ss 3907  df-nul 4275  df-uni 4852  df-tr 5194

This theorem is referenced by:  ttc0elw  36715