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Theorem suceloni 7178
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 5927 . . . . . . . 8 (𝐴 ∈ On → (𝑥𝐴𝑥𝐴))
2 velsn 4337 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3 eqimss 3798 . . . . . . . . . 10 (𝑥 = 𝐴𝑥𝐴)
42, 3sylbi 207 . . . . . . . . 9 (𝑥 ∈ {𝐴} → 𝑥𝐴)
54a1i 11 . . . . . . . 8 (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥𝐴))
61, 5orim12d 919 . . . . . . 7 (𝐴 ∈ On → ((𝑥𝐴𝑥 ∈ {𝐴}) → (𝑥𝐴𝑥𝐴)))
7 df-suc 5890 . . . . . . . . 9 suc 𝐴 = (𝐴 ∪ {𝐴})
87eleq2i 2831 . . . . . . . 8 (𝑥 ∈ suc 𝐴𝑥 ∈ (𝐴 ∪ {𝐴}))
9 elun 3896 . . . . . . . 8 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
108, 9bitr2i 265 . . . . . . 7 ((𝑥𝐴𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11 oridm 537 . . . . . . 7 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
126, 10, 113imtr3g 284 . . . . . 6 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥𝐴))
13 sssucid 5963 . . . . . 6 𝐴 ⊆ suc 𝐴
14 sstr2 3751 . . . . . 6 (𝑥𝐴 → (𝐴 ⊆ suc 𝐴𝑥 ⊆ suc 𝐴))
1512, 13, 14syl6mpi 67 . . . . 5 (𝐴 ∈ On → (𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴))
1615ralrimiv 3103 . . . 4 (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17 dftr3 4908 . . . 4 (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
1816, 17sylibr 224 . . 3 (𝐴 ∈ On → Tr suc 𝐴)
19 onss 7155 . . . . 5 (𝐴 ∈ On → 𝐴 ⊆ On)
20 snssi 4484 . . . . 5 (𝐴 ∈ On → {𝐴} ⊆ On)
2119, 20unssd 3932 . . . 4 (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
227, 21syl5eqss 3790 . . 3 (𝐴 ∈ On → suc 𝐴 ⊆ On)
23 ordon 7147 . . . 4 Ord On
24 trssord 5901 . . . . 5 ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25243exp 1113 . . . 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 7175 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ V)
29 elong 5892 . . 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 1632  wcel 2139  wral 3050  Vcvv 3340  cun 3713  wss 3715  {csn 4321  Tr wtr 4904  Ord word 5883  Oncon0 5884  suc csuc 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-suc 5890
This theorem is referenced by:  ordsuc  7179  unon  7196  onsuci  7203  ordunisuc2  7209  ordzsl  7210  onzsl  7211  tfindsg  7225  dfom2  7232  findsg  7258  tfrlem12  7654  oasuc  7773  omsuc  7775  onasuc  7777  oacl  7784  oneo  7830  omeulem1  7831  omeulem2  7832  oeordi  7836  oeworde  7842  oelim2  7844  oelimcl  7849  oeeulem  7850  oeeui  7851  oaabs2  7894  omxpenlem  8226  card2inf  8625  cantnflt  8742  cantnflem1d  8758  cnfcom  8770  r1ordg  8814  bndrank  8877  r1pw  8881  r1pwALT  8882  tcrank  8920  onssnum  9053  dfac12lem2  9158  cfsuc  9271  cfsmolem  9284  fin1a2lem1  9414  fin1a2lem2  9415  ttukeylem7  9529  alephreg  9596  gch2  9689  winainflem  9707  winalim2  9710  r1wunlim  9751  nqereu  9943  noextend  32125  noresle  32152  nosupno  32155  ontgval  32736  ontgsucval  32737  onsuctop  32738  sucneqond  33524  onsetreclem2  42962
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