Theorem ss1o0el1 4332

Description: A subclass of {∅} contains the empty set if and only if it equals {∅}. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)

Assertion

Ref	Expression
ss1o0el1	⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅}))

Proof of Theorem ss1o0el1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2838	. . . 4 ⊢ (∅ ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
2		sssnm 3877	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅}))
3	1, 2	syl 14	. . 3 ⊢ (∅ ∈ 𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅}))
4	3	biimpcd 159	. 2 ⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 → 𝐴 = {∅}))
5		0ex 4258	. . . 4 ⊢ ∅ ∈ V
6	5	snid 3739	. . 3 ⊢ ∅ ∈ {∅}
7		eleq2 2302	. . 3 ⊢ (𝐴 = {∅} → (∅ ∈ 𝐴 ↔ ∅ ∈ {∅}))
8	6, 7	mpbiri 168	. 2 ⊢ (𝐴 = {∅} → ∅ ∈ 𝐴)
9	4, 8	impbid1 142	1 ⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅}))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ⊆ wss 3220  ∅c0 3520  {csn 3708

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4257

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3714

This theorem is referenced by:  exmid01  4333  ss1o0el1o  7214