Theorem rext 5428

Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)

Assertion

Ref	Expression
rext	⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem rext

Step	Hyp	Ref	Expression
1		vsnid 4644	. . 3 ⊢ 𝑥 ∈ {𝑥}
2		vsnex 5409	. . . 4 ⊢ {𝑥} ∈ V
3		eleq2 2824	. . . . 5 ⊢ (𝑧 = {𝑥} → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ {𝑥}))
4		eleq2 2824	. . . . 5 ⊢ (𝑧 = {𝑥} → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ {𝑥}))
5	3, 4	imbi12d 344	. . . 4 ⊢ (𝑧 = {𝑥} → ((𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
6	2, 5	spcv 3589	. . 3 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
7	1, 6	mpi 20	. 2 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑦 ∈ {𝑥})
8		velsn 4622	. . 3 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
9		equcomi 2017	. . 3 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
10	8, 9	sylbi 217	. 2 ⊢ (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
11	7, 10	syl 17	1 ⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)

Syntax hints:   → wi 4  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {csn 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-sn 4607  df-pr 4609

This theorem is referenced by: (None)