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Theorem suceloni 6960
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 5725 . . . . . . . 8 (𝐴 ∈ On → (𝑥𝐴𝑥𝐴))
2 velsn 4164 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 eqimss 3636 . . . . . . . . . 10 (𝑥 = 𝐴𝑥𝐴)
42, 3sylbi 207 . . . . . . . . 9 (𝑥 ∈ {𝐴} → 𝑥𝐴)
54a1i 11 . . . . . . . 8 (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥𝐴))
61, 5orim12d 882 . . . . . . 7 (𝐴 ∈ On → ((𝑥𝐴𝑥 ∈ {𝐴}) → (𝑥𝐴𝑥𝐴)))
7 df-suc 5688 . . . . . . . . 9 suc 𝐴 = (𝐴 ∪ {𝐴})
87eleq2i 2690 . . . . . . . 8 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
9 elun 3731 . . . . . . . 8 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
108, 9bitr2i 265 . . . . . . 7 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11 oridm 536 . . . . . . 7 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
126, 10, 113imtr3g 284 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥𝐴))
13 sssucid 5761 . . . . . 6 𝐴 ⊆ suc 𝐴
14 sstr2 3590 . . . . . 6 (𝑥𝐴 → (𝐴 ⊆ suc 𝐴𝑥 ⊆ suc 𝐴))
1512, 13, 14syl6mpi 67 . . . . 5 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴))
1615ralrimiv 2959 . . . 4 (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17 dftr3 4716 . . . 4 (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
1816, 17sylibr 224 . . 3 (𝐴 ∈ On → Tr suc 𝐴)
19 onss 6937 . . . . 5 (𝐴 ∈ On → 𝐴 ⊆ On)
20 snssi 4308 . . . . 5 (𝐴 ∈ On → {𝐴} ⊆ On)
2119, 20unssd 3767 . . . 4 (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
227, 21syl5eqss 3628 . . 3 (𝐴 ∈ On → suc 𝐴 ⊆ On)
23 ordon 6929 . . . 4 Ord On
24 trssord 5699 . . . . 5 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25243exp 1261 . . . 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 6957 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ V)
29 elong 5690 . . 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 383   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cun 3553  wss 3555  {csn 4148  Tr wtr 4712  Ord word 5681  Oncon0 5682  suc csuc 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686  df-suc 5688
This theorem is referenced by:  ordsuc  6961  unon  6978  onsuci  6985  ordunisuc2  6991  ordzsl  6992  onzsl  6993  tfindsg  7007  dfom2  7014  findsg  7040  tfrlem12  7430  oasuc  7549  omsuc  7551  onasuc  7553  oacl  7560  oneo  7606  omeulem1  7607  omeulem2  7608  oeordi  7612  oeworde  7618  oelim2  7620  oelimcl  7625  oeeulem  7626  oeeui  7627  oaabs2  7670  omxpenlem  8005  card2inf  8404  cantnflt  8513  cantnflem1d  8529  cnfcom  8541  r1ordg  8585  bndrank  8648  r1pw  8652  r1pwALT  8653  tcrank  8691  onssnum  8807  dfac12lem2  8910  cfsuc  9023  cfsmolem  9036  fin1a2lem1  9166  fin1a2lem2  9167  ttukeylem7  9281  alephreg  9348  gch2  9441  winainflem  9459  winalim2  9462  r1wunlim  9503  nqereu  9695  noextend  31524  noreslege  31571  ontgval  32072  ontgsucval  32073  onsuctop  32074  sucneqond  32845  onsetreclem2  41742
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