Theorem eqsnm 3832

Description: Two ways to express that an inhabited set equals a singleton. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
eqsnm	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqsnm

Step	Hyp	Ref	Expression
1		sssnm 3831	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
2		dfss3 3213	. . 3 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵})
3		velsn 3683	. . . 4 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	3	ralbii 2536	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
5	2, 4	bitri 184	. 2 ⊢ (𝐴 ⊆ {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵)
6	1, 5	bitr3di 195	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 = {𝐵} ↔ ∀𝑥 ∈ 𝐴 𝑥 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672

This theorem is referenced by:  01eq0ring  14147  nninfall  16334