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Theorem onunsnss 6386
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 4294 . . . . 5 ¬ 𝐵𝐵
2 elsni 3421 . . . . . . . 8 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
32adantl 266 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥 = 𝐵)
4 simplr 490 . . . . . . 7 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝑥𝐵)
53, 4eqeltrrd 2131 . . . . . 6 ((((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) ∧ 𝑥 ∈ {𝐵}) → 𝐵𝐵)
65ex 112 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥 ∈ {𝐵} → 𝐵𝐵))
71, 6mtoi 600 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → ¬ 𝑥 ∈ {𝐵})
8 snidg 3428 . . . . . . . . 9 (𝐵𝑉𝐵 ∈ {𝐵})
9 elun2 3139 . . . . . . . . 9 (𝐵 ∈ {𝐵} → 𝐵 ∈ (𝐴 ∪ {𝐵}))
108, 9syl 14 . . . . . . . 8 (𝐵𝑉𝐵 ∈ (𝐴 ∪ {𝐵}))
1110adantr 265 . . . . . . 7 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵 ∈ (𝐴 ∪ {𝐵}))
12 ontr1 4154 . . . . . . . 8 ((𝐴 ∪ {𝐵}) ∈ On → ((𝑥𝐵𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
1312adantl 266 . . . . . . 7 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → ((𝑥𝐵𝐵 ∈ (𝐴 ∪ {𝐵})) → 𝑥 ∈ (𝐴 ∪ {𝐵})))
1411, 13mpan2d 412 . . . . . 6 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥 ∈ (𝐴 ∪ {𝐵})))
1514imp 119 . . . . 5 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐴 ∪ {𝐵}))
16 elun 3112 . . . . 5 (𝑥 ∈ (𝐴 ∪ {𝐵}) ↔ (𝑥𝐴𝑥 ∈ {𝐵}))
1715, 16sylib 131 . . . 4 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → (𝑥𝐴𝑥 ∈ {𝐵}))
187, 17ecased 1255 . . 3 (((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) ∧ 𝑥𝐵) → 𝑥𝐴)
1918ex 112 . 2 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → (𝑥𝐵𝑥𝐴))
2019ssrdv 2979 1 ((𝐵𝑉 ∧ (𝐴 ∪ {𝐵}) ∈ On) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639   = wceq 1259  wcel 1409  cun 2943  wss 2945  {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-un 2950  df-in 2952  df-ss 2959  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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