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Theorem ordunisuc2 7837
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 7836 . 2 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2 ralnex 3071 . . 3 (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3 onsuc 7803 . . . . . . . . . 10 (𝑥 ∈ On → suc 𝑥 ∈ On)
4 eloni 6374 . . . . . . . . . 10 (suc 𝑥 ∈ On → Ord suc 𝑥)
53, 4syl 17 . . . . . . . . 9 (𝑥 ∈ On → Ord suc 𝑥)
6 ordtri3 6400 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
75, 6sylan2 592 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
87con2bid 354 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ ¬ 𝐴 = suc 𝑥))
9 onnbtwn 6458 . . . . . . . . . . . . 13 (𝑥 ∈ On → ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
10 imnan 399 . . . . . . . . . . . . 13 ((𝑥𝐴 → ¬ 𝐴 ∈ suc 𝑥) ↔ ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
119, 10sylibr 233 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥𝐴 → ¬ 𝐴 ∈ suc 𝑥))
1211con2d 134 . . . . . . . . . . 11 (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → ¬ 𝑥𝐴))
13 pm2.21 123 . . . . . . . . . . 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 856 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) → (𝑥𝐴 → suc 𝑥𝐴)))
19 eloni 6374 . . . . . . . . . . . . . 14 (𝑥 ∈ On → Ord 𝑥)
20 ordtri2or 6462 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2119, 20sylan 579 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2221ancoms 458 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝑥𝐴𝐴𝑥))
2322orcomd 868 . . . . . . . . . . 11 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝑥𝐴))
2423adantr 480 . . . . . . . . . 10 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝑥𝐴))
25 ordsssuc2 6455 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2625biimpd 228 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2726adantr 480 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝐴 ∈ suc 𝑥))
28 simpr 484 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝑥𝐴 → suc 𝑥𝐴))
2927, 28orim12d 962 . . . . . . . . . 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 412 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝑥𝐴 → suc 𝑥𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
3218, 31impbid 211 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
338, 32bitr3d 281 . . . . . 6 ((Ord 𝐴𝑥 ∈ On) → (¬ 𝐴 = suc 𝑥 ↔ (𝑥𝐴 → suc 𝑥𝐴)))
3433pm5.74da 801 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴))))
35 impexp 450 . . . . . 6 (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)))
36 simpr 484 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝑥𝐴)
37 ordelon 6388 . . . . . . . . . 10 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
3837ex 412 . . . . . . . . 9 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
3938ancrd 551 . . . . . . . 8 (Ord 𝐴 → (𝑥𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
4036, 39impbid2 225 . . . . . . 7 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥𝐴))
4140imbi1d 341 . . . . . 6 (Ord 𝐴 → (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4235, 41bitr3id 285 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4334, 42bitrd 279 . . . 4 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥𝐴 → suc 𝑥𝐴)))
4443ralbidv2 3172 . . 3 (Ord 𝐴 → (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
452, 44bitr3id 285 . 2 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
461, 45bitrd 279 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844   = wceq 1540  wcel 2105  wral 3060  wrex 3069  wss 3948   cuni 4908  Ord word 6363  Oncon0 6364  suc csuc 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by:  dflim4  7841  limsuc2  42086
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