Theorem ss1o0el1 4241

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 2788	. . . 4 ⊢ (∅ ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
2		sssnm 3795	. . . 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 4171	. . . 4 ⊢ ∅ ∈ V
6	5	snid 3664	. . 3 ⊢ ∅ ∈ {∅}
7		eleq2 2269	. . 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 1373  ∃wex 1515   ∈ wcel 2176   ⊆ wss 3166  ∅c0 3460  {csn 3633

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-nul 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639

This theorem is referenced by:  exmid01  4242  ss1o0el1o  7010