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Theorem ss1o0el1 4181
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 ⊆ {∅} → (∅ ∈ 𝐴𝐴 = {∅}))

Proof of Theorem ss1o0el1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex2 2746 . . . 4 (∅ ∈ 𝐴 → ∃𝑥 𝑥𝐴)
2 sssnm 3739 . . . 4 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅}))
31, 2syl 14 . . 3 (∅ ∈ 𝐴 → (𝐴 ⊆ {∅} ↔ 𝐴 = {∅}))
43biimpcd 158 . 2 (𝐴 ⊆ {∅} → (∅ ∈ 𝐴𝐴 = {∅}))
5 0ex 4114 . . . 4 ∅ ∈ V
65snid 3612 . . 3 ∅ ∈ {∅}
7 eleq2 2234 . . 3 (𝐴 = {∅} → (∅ ∈ 𝐴 ↔ ∅ ∈ {∅}))
86, 7mpbiri 167 . 2 (𝐴 = {∅} → ∅ ∈ 𝐴)
94, 8impbid1 141 1 (𝐴 ⊆ {∅} → (∅ ∈ 𝐴𝐴 = {∅}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1348  wex 1485  wcel 2141  wss 3121  c0 3414  {csn 3581
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 4113
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 3587
This theorem is referenced by:  exmid01  4182  ss1o0el1o  6888
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