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Theorem snsssn 4480
Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1 𝐴 ∈ V
Assertion
Ref Expression
snsssn ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Proof of Theorem snsssn
StepHypRef Expression
1 sssn 4466 . 2 ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2 sneqr.1 . . . . . 6 𝐴 ∈ V
32snnz 4415 . . . . 5 {𝐴} ≠ ∅
43neii 2898 . . . 4 ¬ {𝐴} = ∅
54pm2.21i 116 . . 3 ({𝐴} = ∅ → 𝐴 = 𝐵)
62sneqr 4479 . . 3 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
75, 6jaoi 393 . 2 (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
81, 7sylbi 207 1 ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1596  wcel 2103  Vcvv 3304  wss 3680  c0 4023  {csn 4285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694  df-nul 4024  df-sn 4286
This theorem is referenced by:  k0004lem3  38866
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