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Theorem onunsnss 6892
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 4523 . . . . 5 ¬ 𝐵𝐵
2 elsni 3599 . . . . . . . 8 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
32adantl 275 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 = 𝐵)
4 simplr 525 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥𝐵)
53, 4eqeltrrd 2248 . . . . . 6 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝐵𝐵)
65ex 114 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥 ∈ {𝐵} → 𝐵𝐵))
71, 6mtoi 659 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → ¬ 𝑥 ∈ {𝐵})
8 snidg 3610 . . . . . . . . 9 (𝐵𝑉𝐵 ∈ {𝐵})
9 elun2 3295 . . . . . . . . 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 4372 . . . . . . . 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 426 . . . . . 6 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥 ∈ (𝐴 ∪ {𝐵})))
1514imp 123 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
16 elun 3268 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
1715, 16sylib 121 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥𝐴𝑥 ∈ {𝐵}))
187, 17ecased 1344 . . 3 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥𝐴)
1918ex 114 . 2 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥𝐴))
2019ssrdv 3153 1 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 703   = wceq 1348  wcel 2141  cun 3119  wss 3121  {csn 3581  Oncon0 4346
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351
This theorem is referenced by: (None)
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