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Theorem onunsnss 6808
 Description: Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
Assertion
Ref Expression
onunsnss ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)

Proof of Theorem onunsnss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elirr 4459 . . . . 5 ¬ 𝐵𝐵
2 elsni 3545 . . . . . . . 8 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
32adantl 275 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 = 𝐵)
4 simplr 519 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥𝐵)
53, 4eqeltrrd 2217 . . . . . 6 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝐵𝐵)
65ex 114 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥 ∈ {𝐵} → 𝐵𝐵))
71, 6mtoi 653 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → ¬ 𝑥 ∈ {𝐵})
8 snidg 3556 . . . . . . . . 9 (𝐵𝑉𝐵 ∈ {𝐵})
9 elun2 3244 . . . . . . . . 9 (𝐵 ∈ {𝐵} → 𝐵 ∈ (𝐴 ∪ {𝐵}))
108, 9syl 14 . . . . . . . 8 (𝐵𝑉𝐵 ∈ (𝐴 ∪ {𝐵}))
1110adantr 274 . . . . . . 7 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
12 ontr1 4314 . . . . . . . 8 ((𝐴 ∪ {𝐵}) ∈ On → ((𝑥𝐵𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
1312adantl 275 . . . . . . 7 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → ((𝑥𝐵𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
1411, 13mpan2d 424 . . . . . 6 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥 ∈ (𝐴 ∪ {𝐵})))
1514imp 123 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
16 elun 3217 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
1715, 16sylib 121 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥𝐴𝑥 ∈ {𝐵}))
187, 17ecased 1327 . . 3 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥𝐴)
1918ex 114 . 2 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥𝐴))
2019ssrdv 3103 1 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697   = wceq 1331   ∈ wcel 1480   ∪ cun 3069   ⊆ wss 3071  {csn 3527  Oncon0 4288 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4455 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-uni 3740  df-tr 4030  df-iord 4291  df-on 4293 This theorem is referenced by: (None)
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