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Theorem suceloni 6905
Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
suceloni (𝐴 ∈ On → suc 𝐴 ∈ On)

Proof of Theorem suceloni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 onelss 5683 . . . . . . . 8 (𝐴 ∈ On → (𝑥𝐴𝑥𝐴))
2 velsn 4141 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 eqimss 3620 . . . . . . . . . 10 (𝑥 = 𝐴𝑥𝐴)
42, 3sylbi 206 . . . . . . . . 9 (𝑥 ∈ {𝐴} → 𝑥𝐴)
54a1i 11 . . . . . . . 8 (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥𝐴))
61, 5orim12d 879 . . . . . . 7 (𝐴 ∈ On → ((𝑥𝐴𝑥 ∈ {𝐴}) → (𝑥𝐴𝑥𝐴)))
7 df-suc 5646 . . . . . . . . 9 suc 𝐴 = (𝐴 ∪ {𝐴})
87eleq2i 2680 . . . . . . . 8 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
9 elun 3715 . . . . . . . 8 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
108, 9bitr2i 264 . . . . . . 7 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11 oridm 535 . . . . . . 7 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
126, 10, 113imtr3g 283 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥𝐴))
13 sssucid 5719 . . . . . 6 𝐴 ⊆ suc 𝐴
14 sstr2 3575 . . . . . 6 (𝑥𝐴 → (𝐴 ⊆ suc 𝐴𝑥 ⊆ suc 𝐴))
1512, 13, 14syl6mpi 65 . . . . 5 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴))
1615ralrimiv 2948 . . . 4 (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17 dftr3 4684 . . . 4 (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
1816, 17sylibr 223 . . 3 (𝐴 ∈ On → Tr suc 𝐴)
19 onss 6882 . . . . 5 (𝐴 ∈ On → 𝐴 ⊆ On)
20 snssi 4280 . . . . 5 (𝐴 ∈ On → {𝐴} ⊆ On)
2119, 20unssd 3751 . . . 4 (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
227, 21syl5eqss 3612 . . 3 (𝐴 ∈ On → suc 𝐴 ⊆ On)
23 ordon 6874 . . . 4 Ord On
24 trssord 5657 . . . . 5 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25243exp 1256 . . . 4 (Tr suc 𝐴 → (suc 𝐴 ⊆ On → (Ord On → Ord suc 𝐴)))
2623, 25mpii 45 . . 3 (Tr suc 𝐴 → (suc 𝐴 ⊆ On → Ord suc 𝐴))
2718, 22, 26sylc 63 . 2 (𝐴 ∈ On → Ord suc 𝐴)
28 sucexg 6902 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ V)
29 elong 5648 . . 3 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3028, 29syl 17 . 2 (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3127, 30mpbird 246 1 (𝐴 ∈ On → suc 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cun 3538  wss 3540  {csn 4125  Tr wtr 4680  Ord word 5639  Oncon0 5640  suc csuc 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646
This theorem is referenced by:  ordsuc  6906  unon  6923  onsuci  6930  ordunisuc2  6936  ordzsl  6937  onzsl  6938  tfindsg  6952  dfom2  6959  findsg  6985  tfrlem12  7372  oasuc  7491  omsuc  7493  onasuc  7495  oacl  7502  oneo  7548  omeulem1  7549  omeulem2  7550  oeordi  7554  oeworde  7560  oelim2  7562  oelimcl  7567  oeeulem  7568  oeeui  7569  oaabs2  7612  omxpenlem  7946  card2inf  8343  cantnflt  8452  cantnflem1d  8468  cnfcom  8480  r1ordg  8524  bndrank  8587  r1pw  8591  r1pwALT  8592  tcrank  8630  onssnum  8746  dfac12lem2  8849  cfsuc  8962  cfsmolem  8975  fin1a2lem1  9105  fin1a2lem2  9106  ttukeylem7  9220  alephreg  9283  gch2  9376  winainflem  9394  winalim2  9397  r1wunlim  9438  nqereu  9630  ontgval  31600  ontgsucval  31601  onsuctop  31602  sucneqond  32389  onsetreclem2  42248
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