Theorem issn 4790

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 2038	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2	1	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑥))
3	2	ralbidv 3185	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥))
4		ne0i 4293	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
5		eqsn 4787	. . . . 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 284	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ 𝐴 = {𝑥}))
8		sneq 4592	. . . . 5 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
9	8	eqeq2d 2773	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥}))
10	9	spcegv 3556	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧}))
11	7, 10	sylbid 242	. 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}))
12	11	rexlimiv 3156	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560  ∃wex 1799   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086  ∅c0 4285  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286  df-sn 4583

This theorem is referenced by:  n0snor2el  4791  f1cdmsn  7266