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Theorem snon0 6356
 Description: An ordinal which is a singleton is {∅}. (Contributed by Jim Kingdon, 19-Oct-2021.)
Assertion
Ref Expression
snon0 ((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)

Proof of Theorem snon0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elirr 4266 . . 3 ¬ 𝐴𝐴
2 snidg 3400 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
32adantr 261 . . . . . 6 ((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 ∈ {𝐴})
4 ontr1 4126 . . . . . . 7 ({𝐴} ∈ On → ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
54adantl 262 . . . . . 6 ((𝐴𝑉 ∧ {𝐴} ∈ On) → ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
63, 5mpan2d 404 . . . . 5 ((𝐴𝑉 ∧ {𝐴} ∈ On) → (𝑥𝐴𝑥 ∈ {𝐴}))
7 elsni 3393 . . . . 5 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
86, 7syl6 29 . . . 4 ((𝐴𝑉 ∧ {𝐴} ∈ On) → (𝑥𝐴𝑥 = 𝐴))
9 eleq1 2100 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
109biimpcd 148 . . . 4 (𝑥𝐴 → (𝑥 = 𝐴𝐴𝐴))
118, 10sylcom 25 . . 3 ((𝐴𝑉 ∧ {𝐴} ∈ On) → (𝑥𝐴𝐴𝐴))
121, 11mtoi 590 . 2 ((𝐴𝑉 ∧ {𝐴} ∈ On) → ¬ 𝑥𝐴)
1312eq0rdv 3261 1 ((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∅c0 3224  {csn 3375  Oncon0 4100 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by: (None)
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