Theorem ss1o0el1 4287

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 2819	. . . 4 ⊢ (∅ ∈ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
2		sssnm 3837	. . . 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 4216	. . . 4 ⊢ ∅ ∈ V
6	5	snid 3700	. . 3 ⊢ ∅ ∈ {∅}
7		eleq2 2295	. . 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 1397  ∃wex 1540   ∈ wcel 2202   ⊆ wss 3200  ∅c0 3494  {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675

This theorem is referenced by:  exmid01  4288  ss1o0el1o  7105