Theorem issn 4363

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 group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem issn

Step	Hyp	Ref	Expression
1		equcom 1945	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2	1	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = 𝑦 ↔ 𝑦 = 𝑥))
3	2	ralbidv 2986	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑦 ∈ 𝐴 𝑦 = 𝑥))
4		ne0i 3921	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
5		eqsn 4361	. . . . 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 271	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ 𝐴 = {𝑥}))
8		sneq 4187	. . . . 5 ⊢ (𝑧 = 𝑥 → {𝑧} = {𝑥})
9	8	eqeq2d 2632	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐴 = {𝑧} ↔ 𝐴 = {𝑥}))
10	9	spcegv 3294	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 = {𝑥} → ∃𝑧 𝐴 = {𝑧}))
11	7, 10	sylbid 230	. 2 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧}))
12	11	rexlimiv 3027	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 → ∃𝑧 𝐴 = {𝑧})

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  ∅c0 3915  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by:  n0snor2el  4364