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Theorem suceloni 7055
 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 5804 . . . . . . . 8 (𝐴 ∈ On → (𝑥𝐴𝑥𝐴))
2 velsn 4226 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 eqimss 3690 . . . . . . . . . 10 (𝑥 = 𝐴𝑥𝐴)
42, 3sylbi 207 . . . . . . . . 9 (𝑥 ∈ {𝐴} → 𝑥𝐴)
54a1i 11 . . . . . . . 8 (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥𝐴))
61, 5orim12d 901 . . . . . . 7 (𝐴 ∈ On → ((𝑥𝐴𝑥 ∈ {𝐴}) → (𝑥𝐴𝑥𝐴)))
7 df-suc 5767 . . . . . . . . 9 suc 𝐴 = (𝐴 ∪ {𝐴})
87eleq2i 2722 . . . . . . . 8 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
9 elun 3786 . . . . . . . 8 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
108, 9bitr2i 265 . . . . . . 7 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11 oridm 535 . . . . . . 7 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
126, 10, 113imtr3g 284 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥𝐴))
13 sssucid 5840 . . . . . 6 𝐴 ⊆ suc 𝐴
14 sstr2 3643 . . . . . 6 (𝑥𝐴 → (𝐴 ⊆ suc 𝐴𝑥 ⊆ suc 𝐴))
1512, 13, 14syl6mpi 67 . . . . 5 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴))
1615ralrimiv 2994 . . . 4 (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17 dftr3 4789 . . . 4 (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
1816, 17sylibr 224 . . 3 (𝐴 ∈ On → Tr suc 𝐴)
19 onss 7032 . . . . 5 (𝐴 ∈ On → 𝐴 ⊆ On)
20 snssi 4371 . . . . 5 (𝐴 ∈ On → {𝐴} ⊆ On)
2119, 20unssd 3822 . . . 4 (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
227, 21syl5eqss 3682 . . 3 (𝐴 ∈ On → suc 𝐴 ⊆ On)
23 ordon 7024 . . . 4 Ord On
24 trssord 5778 . . . . 5 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25243exp 1283 . . . 4 (Tr suc 𝐴 → (suc 𝐴 ⊆ On → (Ord On → Ord suc 𝐴)))
2623, 25mpii 46 . . 3 (Tr suc 𝐴 → (suc 𝐴 ⊆ On → Ord suc 𝐴))
2718, 22, 26sylc 65 . 2 (𝐴 ∈ On → Ord suc 𝐴)
28 sucexg 7052 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ V)
29 elong 5769 . . 3 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3028, 29syl 17 . 2 (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
3127, 30mpbird 247 1 (𝐴 ∈ On → suc 𝐴 ∈ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  {csn 4210  Tr wtr 4785  Ord word 5760  Oncon0 5761  suc csuc 5763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767 This theorem is referenced by:  ordsuc  7056  unon  7073  onsuci  7080  ordunisuc2  7086  ordzsl  7087  onzsl  7088  tfindsg  7102  dfom2  7109  findsg  7135  tfrlem12  7530  oasuc  7649  omsuc  7651  onasuc  7653  oacl  7660  oneo  7706  omeulem1  7707  omeulem2  7708  oeordi  7712  oeworde  7718  oelim2  7720  oelimcl  7725  oeeulem  7726  oeeui  7727  oaabs2  7770  omxpenlem  8102  card2inf  8501  cantnflt  8607  cantnflem1d  8623  cnfcom  8635  r1ordg  8679  bndrank  8742  r1pw  8746  r1pwALT  8747  tcrank  8785  onssnum  8901  dfac12lem2  9004  cfsuc  9117  cfsmolem  9130  fin1a2lem1  9260  fin1a2lem2  9261  ttukeylem7  9375  alephreg  9442  gch2  9535  winainflem  9553  winalim2  9556  r1wunlim  9597  nqereu  9789  noextend  31944  noresle  31971  nosupno  31974  ontgval  32555  ontgsucval  32556  onsuctop  32557  sucneqond  33343  onsetreclem2  42777
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