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Theorem snssb 3727
Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
Assertion
Ref Expression
snssb ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴𝐵))

Proof of Theorem snssb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3146 . 2 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
2 velsn 3611 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
32imbi1i 238 . . 3 ((𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ (𝑥 = 𝐴𝑥𝐵))
43albii 1470 . 2 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
5 eleq1 2240 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
65pm5.74i 180 . . . 4 ((𝑥 = 𝐴𝑥𝐵) ↔ (𝑥 = 𝐴𝐴𝐵))
76albii 1470 . . 3 (∀𝑥(𝑥 = 𝐴𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝐴𝐵))
8 19.23v 1883 . . 3 (∀𝑥(𝑥 = 𝐴𝐴𝐵) ↔ (∃𝑥 𝑥 = 𝐴𝐴𝐵))
9 isset 2745 . . . . 5 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
109bicomi 132 . . . 4 (∃𝑥 𝑥 = 𝐴𝐴 ∈ V)
1110imbi1i 238 . . 3 ((∃𝑥 𝑥 = 𝐴𝐴𝐵) ↔ (𝐴 ∈ V → 𝐴𝐵))
127, 8, 113bitri 206 . 2 (∀𝑥(𝑥 = 𝐴𝑥𝐵) ↔ (𝐴 ∈ V → 𝐴𝐵))
131, 4, 123bitri 206 1 ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  Vcvv 2739  wss 3131  {csn 3594
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-sn 3600
This theorem is referenced by:  snssg  3728
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