Theorem ss1o0el1 4183

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 2746	. . . 4 ⊢ (∅ ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
2		sssnm 3741	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅}))
3	1, 2	syl 14	. . 3 ⊢ (∅ ∈ 𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅}))
4	3	biimpcd 158	. 2 ⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 → 𝐴 = {∅}))
5		0ex 4116	. . . 4 ⊢ ∅ ∈ V
6	5	snid 3614	. . 3 ⊢ ∅ ∈ {∅}
7		eleq2 2234	. . 3 ⊢ (𝐴 = {∅} → (∅ ∈ 𝐴 ↔ ∅ ∈ {∅}))
8	6, 7	mpbiri 167	. 2 ⊢ (𝐴 = {∅} → ∅ ∈ 𝐴)
9	4, 8	impbid1 141	1 ⊢ (𝐴 ⊆ {∅} → (∅ ∈ 𝐴 ↔ 𝐴 = {∅}))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141   ⊆ wss 3121  ∅c0 3414  {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589

This theorem is referenced by:  exmid01  4184  ss1o0el1o  6890