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Theorem onunsnss 6915
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 4540 . . . . 5 ¬ 𝐵𝐵
2 elsni 3610 . . . . . . . 8 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
32adantl 277 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 = 𝐵)
4 simplr 528 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥𝐵)
53, 4eqeltrrd 2255 . . . . . 6 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝐵𝐵)
65ex 115 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥 ∈ {𝐵} → 𝐵𝐵))
71, 6mtoi 664 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → ¬ 𝑥 ∈ {𝐵})
8 snidg 3621 . . . . . . . . 9 (𝐵𝑉𝐵 ∈ {𝐵})
9 elun2 3303 . . . . . . . . 9 (𝐵 ∈ {𝐵} → 𝐵 ∈ (𝐴 ∪ {𝐵}))
108, 9syl 14 . . . . . . . 8 (𝐵𝑉𝐵 ∈ (𝐴 ∪ {𝐵}))
1110adantr 276 . . . . . . 7 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
12 ontr1 4389 . . . . . . . 8 ((𝐴 ∪ {𝐵}) ∈ On → ((𝑥𝐵𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
1312adantl 277 . . . . . . 7 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → ((𝑥𝐵𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
1411, 13mpan2d 428 . . . . . 6 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥 ∈ (𝐴 ∪ {𝐵})))
1514imp 124 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
16 elun 3276 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
1715, 16sylib 122 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥𝐴𝑥 ∈ {𝐵}))
187, 17ecased 1349 . . 3 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥𝐴)
1918ex 115 . 2 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥𝐴))
2019ssrdv 3161 1 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708   = wceq 1353  wcel 2148  cun 3127  wss 3129  {csn 3592  Oncon0 4363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-uni 3810  df-tr 4102  df-iord 4366  df-on 4368
This theorem is referenced by: (None)
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