Theorem issn 4775

Description: A sufficient condition for a (nonempty) set to be a singleton. (Contributed by AV, 20-Sep-2020.)

Assertion

Ref	Expression
issn	⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})

Distinct variable groups:   𝑥,𝐴,𝑦   𝑧,𝐴,𝑥

Proof of Theorem issn

Step	Hyp	Ref	Expression
1		equcom 2020	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2	1	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑥))
3	2	ralbidv 3160	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥))
4		ne0i 4281	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
5		eqsn 4772	. . . . 5 ⊢ (𝐴 ≠ ∅ → (𝐴 = {𝑥} ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥))
6	4, 5	syl 17	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥))
7	3, 6	bitr4d 282	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ 𝐴 = {𝑥}))
8		sneq 4577	. . . . 5 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
9	8	eqeq2d 2747	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥}))
10	9	spcegv 3539	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧}))
11	7, 10	sylbid 240	. 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}))
12	11	rexlimiv 3131	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  ∅c0 4273  {csn 4567

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-v 3431  df-dif 3892  df-ss 3906  df-nul 4274  df-sn 4568

This theorem is referenced by:  n0snor2el  4776  f1cdmsn  7237