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Theorem snon0 6387
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 4294 . . 3 ¬ 𝐴𝐴
2 snidg 3428 . . . . . . 7 (𝐴𝑉𝐴 ∈ {𝐴})
32adantr 265 . . . . . 6 ((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 ∈ {𝐴})
4 ontr1 4154 . . . . . . 7 ({𝐴} ∈ On → ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
54adantl 266 . . . . . 6 ((𝐴𝑉 ∧ {𝐴} ∈ On) → ((𝑥𝐴𝐴 ∈ {𝐴}) → 𝑥 ∈ {𝐴}))
63, 5mpan2d 412 . . . . 5 ((𝐴𝑉 ∧ {𝐴} ∈ On) → (𝑥𝐴𝑥 ∈ {𝐴}))
7 elsni 3421 . . . . 5 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
86, 7syl6 33 . . . 4 ((𝐴𝑉 ∧ {𝐴} ∈ On) → (𝑥𝐴𝑥 = 𝐴))
9 eleq1 2116 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
109biimpcd 152 . . . 4 (𝑥𝐴 → (𝑥 = 𝐴𝐴𝐴))
118, 10sylcom 28 . . 3 ((𝐴𝑉 ∧ {𝐴} ∈ On) → (𝑥𝐴𝐴𝐴))
121, 11mtoi 600 . 2 ((𝐴𝑉 ∧ {𝐴} ∈ On) → ¬ 𝑥𝐴)
1312eq0rdv 3289 1 ((𝐴𝑉 ∧ {𝐴} ∈ On) → 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  c0 3252  {csn 3403  Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133
This theorem is referenced by: (None)
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