Theorem ss1o0el1 4281

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 2816	. . . 4 ⊢ (∅ ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
2		sssnm 3832	. . . 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 4211	. . . 4 ⊢ ∅ ∈ V
6	5	snid 3697	. . 3 ⊢ ∅ ∈ {∅}
7		eleq2 2293	. . 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 1395  ∃wex 1538   ∈ wcel 2200   ⊆ wss 3197  ∅c0 3491  {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-in1 617  ax-in2 618  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  ax-nul 4210

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-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672

This theorem is referenced by:  exmid01  4282  ss1o0el1o  7075