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Theorem ordunisuc2 7785
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 7784 . 2 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2 ralnex 3076 . . 3 (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3 onsuc 7751 . . . . . . . . . 10 (𝑥 ∈ On → suc 𝑥 ∈ On)
4 eloni 6332 . . . . . . . . . 10 (suc 𝑥 ∈ On → Ord suc 𝑥)
53, 4syl 17 . . . . . . . . 9 (𝑥 ∈ On → Ord suc 𝑥)
6 ordtri3 6358 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
75, 6sylan2 594 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴)))
87con2bid 355 . . . . . . 7 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) ↔ ¬ 𝐴 = suc 𝑥))
9 onnbtwn 6416 . . . . . . . . . . . . 13 (𝑥 ∈ On → ¬ (𝑥𝐴𝐴 ∈ suc 𝑥))
10 imnan 401 . . . . . . . . . . . . 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 483 . . . . . . . . 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 858 . . . . . . . 8 ((Ord 𝐴𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥𝐴) → (𝑥𝐴 → suc 𝑥𝐴)))
19 eloni 6332 . . . . . . . . . . . . . 14 (𝑥 ∈ On → Ord 𝑥)
20 ordtri2or 6420 . . . . . . . . . . . . . 14 ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2119, 20sylan 581 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥𝐴𝐴𝑥))
2221ancoms 460 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝑥𝐴𝐴𝑥))
2322orcomd 870 . . . . . . . . . . 11 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝑥𝐴))
2423adantr 482 . . . . . . . . . 10 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝑥𝐴))
25 ordsssuc2 6413 . . . . . . . . . . . . 13 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2625biimpd 228 . . . . . . . . . . . 12 ((Ord 𝐴𝑥 ∈ On) → (𝐴𝑥𝐴 ∈ suc 𝑥))
2726adantr 482 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝐴𝑥𝐴 ∈ suc 𝑥))
28 simpr 486 . . . . . . . . . . 11 (((Ord 𝐴𝑥 ∈ On) ∧ (𝑥𝐴 → suc 𝑥𝐴)) → (𝑥𝐴 → suc 𝑥𝐴))
2927, 28orim12d 964 . . . . . . . . . 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 414 . . . . . . . 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 803 . . . . 5 (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴))))
35 impexp 452 . . . . . 6 (((𝑥 ∈ On ∧ 𝑥𝐴) → suc 𝑥𝐴) ↔ (𝑥 ∈ On → (𝑥𝐴 → suc 𝑥𝐴)))
36 simpr 486 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝑥𝐴)
37 ordelon 6346 . . . . . . . . . 10 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
3837ex 414 . . . . . . . . 9 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
3938ancrd 553 . . . . . . . 8 (Ord 𝐴 → (𝑥𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
4036, 39impbid2 225 . . . . . . 7 (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥𝐴) ↔ 𝑥𝐴))
4140imbi1d 342 . . . . . 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 3171 . . 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 397  wo 846   = wceq 1542  wcel 2107  wral 3065  wrex 3074  wss 3915   cuni 4870  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  dflim4  7789  limsuc2  41397
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