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Theorem orduninsuc 7551
 Description: An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
orduninsuc (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem orduninsuc
StepHypRef Expression
1 ordeleqon 7496 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 id 22 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3 unieq 4815 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
42, 3eqeq12d 2814 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅)))
5 eqeq1 2802 . . . . . . 7 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
65rexbidv 3257 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
76notbid 321 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
84, 7bibi12d 349 . . . 4 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)))
9 0elon 6219 . . . . . 6 ∅ ∈ On
109elimel 4495 . . . . 5 if(𝐴 ∈ On, 𝐴, ∅) ∈ On
1110onuninsuci 7548 . . . 4 (if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
128, 11dedth 4484 . . 3 (𝐴 ∈ On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13 unon 7539 . . . . . 6 On = On
1413eqcomi 2807 . . . . 5 On = On
15 onprc 7492 . . . . . . . 8 ¬ On ∈ V
16 vex 3445 . . . . . . . . . 10 𝑥 ∈ V
1716sucex 7519 . . . . . . . . 9 suc 𝑥 ∈ V
18 eleq1 2877 . . . . . . . . 9 (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
1917, 18mpbiri 261 . . . . . . . 8 (On = suc 𝑥 → On ∈ V)
2015, 19mto 200 . . . . . . 7 ¬ On = suc 𝑥
2120a1i 11 . . . . . 6 (𝑥 ∈ On → ¬ On = suc 𝑥)
2221nrex 3229 . . . . 5 ¬ ∃𝑥 ∈ On On = suc 𝑥
2314, 222th 267 . . . 4 (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24 id 22 . . . . . 6 (𝐴 = On → 𝐴 = On)
25 unieq 4815 . . . . . 6 (𝐴 = On → 𝐴 = On)
2624, 25eqeq12d 2814 . . . . 5 (𝐴 = On → (𝐴 = 𝐴 ↔ On = On))
27 eqeq1 2802 . . . . . . 7 (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
2827rexbidv 3257 . . . . . 6 (𝐴 = On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On On = suc 𝑥))
2928notbid 321 . . . . 5 (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
3026, 29bibi12d 349 . . . 4 (𝐴 = On → ((𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
3123, 30mpbiri 261 . . 3 (𝐴 = On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
3212, 31jaoi 854 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
331, 32sylbi 220 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3442  ∅c0 4246  ifcif 4428  ∪ cuni 4804  Ord word 6165  Oncon0 6166  suc csuc 6168 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-tr 5141  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6169  df-on 6170  df-suc 6172 This theorem is referenced by:  ordunisuc2  7552  ordzsl  7553  dflim3  7555  nnsuc  7590
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