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Theorem ordunisuc2 7086
Description: An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
ordunisuc2 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ordunisuc2
StepHypRef Expression
1 orduninsuc 7085 . 2 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2 ralnex 3021 . . 3 (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3 suceloni 7055 . . . . . . . . . 10 (𝑥 ∈ On → suc 𝑥 ∈ On)
4 eloni 5771 . . . . . . . . . 10 (suc 𝑥 ∈ On → Ord suc 𝑥)
53, 4syl 17 . . . . . . . . 9 (𝑥 ∈ On → Ord suc 𝑥)
6 ordtri3 5797 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
75, 6sylan2 490 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
87con2bid 343 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ ¬ 𝐴 = suc 𝑥))
9 onnbtwn 5856 . . . . . . . . . . . . 13 (𝑥 ∈ On → ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
10 imnan 437 . . . . . . . . . . . . 13 ((𝑥𝐴 → ¬ 𝐴 ∈ suc 𝑥) ↔ ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
119, 10sylibr 224 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥𝐴 → ¬ 𝐴 ∈ suc 𝑥))
1211con2d 129 . . . . . . . . . . 11 (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → ¬ 𝑥𝐴))
13 pm2.21 120 . . . . . . . . . . 11 𝑥𝐴 → (𝑥𝐴 → suc 𝑥𝐴))
1412, 13syl6 35 . . . . . . . . . 10 (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → (𝑥𝐴 → suc 𝑥𝐴)))
1514adantl 481 . . . . . . . . 9 ((Ord 𝐴𝑥 ∈ On) → (𝐴 ∈ suc 𝑥 → (𝑥𝐴 → suc 𝑥𝐴)))
16 ax-1 6 . . . . . . . . . 10 (suc 𝑥𝐴 → (𝑥𝐴 → suc 𝑥𝐴))
1716a1i 11 . . . . . . . . 9 ((Ord 𝐴𝑥 ∈ On) → (suc 𝑥𝐴 → (𝑥𝐴 → suc 𝑥𝐴)))
1815, 17jaod 394 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) → (𝑥𝐴 → suc 𝑥𝐴)))
19 eloni 5771 . . . . . . . . . . . . . 14 (𝑥 ∈ On → Ord 𝑥)
20 ordtri2or 5860 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2119, 20sylan 487 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2221ancoms 468 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝑥𝐴𝐴𝑥))
2322orcomd 402 . . . . . . . . . . 11 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝑥𝐴))
2423adantr 480 . . . . . . . . . 10 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝑥𝐴))
25 ordsssuc2 5852 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2625biimpd 219 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2726adantr 480 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝐴 ∈ suc 𝑥))
28 simpr 476 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝑥𝐴 → suc 𝑥𝐴))
2927, 28orim12d 901 . . . . . . . . . 10 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → ((𝐴𝑥𝑥𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
3024, 29mpd 15 . . . . . . . . 9 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴))
3130ex 449 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝑥𝐴 → suc 𝑥𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
3218, 31impbid 202 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
338, 32bitr3d 270 . . . . . 6 ((Ord 𝐴𝑥 ∈ On) → (¬ 𝐴 = suc 𝑥 ↔ (𝑥𝐴 → suc 𝑥𝐴)))
3433pm5.74da 723 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴))))
35 impexp 461 . . . . . 6 (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)))
36 simpr 476 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝑥𝐴)
37 ordelon 5785 . . . . . . . . . 10 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
3837ex 449 . . . . . . . . 9 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
3938ancrd 576 . . . . . . . 8 (Ord 𝐴 → (𝑥𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
4036, 39impbid2 216 . . . . . . 7 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥𝐴))
4140imbi1d 330 . . . . . 6 (Ord 𝐴 → (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4235, 41syl5bbr 274 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4334, 42bitrd 268 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4443ralbidv2 3013 . . 3 (Ord 𝐴 → (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
452, 44syl5bbr 274 . 2 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
461, 45bitrd 268 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607   cuni 4468  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-pw 4193  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:  dflim4  7090  limsuc2  37928
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