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Theorem ordsuc 4446
Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
Assertion
Ref Expression
ordsuc (Ord 𝐴 ↔ Ord suc 𝐴)

Proof of Theorem ordsuc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordsucim 4384 . 2 (Ord 𝐴 → Ord suc 𝐴)
2 en2lp 4437 . . . . . . . . . 10 ¬ (𝑥𝐴𝐴𝑥)
3 eleq1 2178 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
43biimpac 294 . . . . . . . . . . . 12 ((𝑦𝑥𝑦 = 𝐴) → 𝐴𝑥)
54anim2i 337 . . . . . . . . . . 11 ((𝑥𝐴 ∧ (𝑦𝑥𝑦 = 𝐴)) → (𝑥𝐴𝐴𝑥))
65expr 370 . . . . . . . . . 10 ((𝑥𝐴𝑦𝑥) → (𝑦 = 𝐴 → (𝑥𝐴𝐴𝑥)))
72, 6mtoi 636 . . . . . . . . 9 ((𝑥𝐴𝑦𝑥) → ¬ 𝑦 = 𝐴)
87adantl 273 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → ¬ 𝑦 = 𝐴)
9 elelsuc 4299 . . . . . . . . . . . . . . 15 (𝑥𝐴𝑥 ∈ suc 𝐴)
109adantr 272 . . . . . . . . . . . . . 14 ((𝑥𝐴𝑦𝑥) → 𝑥 ∈ suc 𝐴)
11 ordelss 4269 . . . . . . . . . . . . . 14 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ⊆ suc 𝐴)
1210, 11sylan2 282 . . . . . . . . . . . . 13 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑥 ⊆ suc 𝐴)
1312sseld 3064 . . . . . . . . . . . 12 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1413expr 370 . . . . . . . . . . 11 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥 → (𝑦𝑥𝑦 ∈ suc 𝐴)))
1514pm2.43d 50 . . . . . . . . . 10 ((Ord suc 𝐴𝑥𝐴) → (𝑦𝑥𝑦 ∈ suc 𝐴))
1615impr 374 . . . . . . . . 9 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦 ∈ suc 𝐴)
17 elsuci 4293 . . . . . . . . 9 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1816, 17syl 14 . . . . . . . 8 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → (𝑦𝐴𝑦 = 𝐴))
198, 18ecased 1310 . . . . . . 7 ((Ord suc 𝐴 ∧ (𝑥𝐴𝑦𝑥)) → 𝑦𝐴)
2019ancom2s 538 . . . . . 6 ((Ord suc 𝐴 ∧ (𝑦𝑥𝑥𝐴)) → 𝑦𝐴)
2120ex 114 . . . . 5 (Ord suc 𝐴 → ((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2221alrimivv 1829 . . . 4 (Ord suc 𝐴 → ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
23 dftr2 3996 . . . 4 (Tr 𝐴 ↔ ∀𝑦𝑥((𝑦𝑥𝑥𝐴) → 𝑦𝐴))
2422, 23sylibr 133 . . 3 (Ord suc 𝐴 → Tr 𝐴)
25 sssucid 4305 . . . 4 𝐴 ⊆ suc 𝐴
26 trssord 4270 . . . 4 ((Tr 𝐴𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2725, 26mp3an2 1286 . . 3 ((Tr 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
2824, 27mpancom 416 . 2 (Ord suc 𝐴 → Ord 𝐴)
291, 28impbii 125 1 (Ord 𝐴 ↔ Ord suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680  wal 1312   = wceq 1314  wcel 1463  wss 3039  Tr wtr 3994  Ord word 4252  suc csuc 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-uni 3705  df-tr 3995  df-iord 4256  df-suc 4261
This theorem is referenced by:  nlimsucg  4449  ordpwsucss  4450
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