MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  orduninsuc Structured version   Visualization version   GIF version

Theorem orduninsuc 7189
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 7134 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 id 22 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3 unieq 4580 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
42, 3eqeq12d 2785 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅)))
5 eqeq1 2774 . . . . . . 7 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
65rexbidv 3199 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
76notbid 307 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
84, 7bibi12d 334 . . . 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 5921 . . . . . 6 ∅ ∈ On
109elimel 4287 . . . . 5 if(𝐴 ∈ On, 𝐴, ∅) ∈ On
1110onuninsuci 7186 . . . 4 (if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
128, 11dedth 4276 . . 3 (𝐴 ∈ On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13 unon 7177 . . . . . 6 On = On
1413eqcomi 2779 . . . . 5 On = On
15 onprc 7130 . . . . . . . 8 ¬ On ∈ V
16 vex 3352 . . . . . . . . . 10 𝑥 ∈ V
1716sucex 7157 . . . . . . . . 9 suc 𝑥 ∈ V
18 eleq1 2837 . . . . . . . . 9 (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
1917, 18mpbiri 248 . . . . . . . 8 (On = suc 𝑥 → On ∈ V)
2015, 19mto 188 . . . . . . 7 ¬ On = suc 𝑥
2120a1i 11 . . . . . 6 (𝑥 ∈ On → ¬ On = suc 𝑥)
2221nrex 3147 . . . . 5 ¬ ∃𝑥 ∈ On On = suc 𝑥
2314, 222th 254 . . . 4 (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24 id 22 . . . . . 6 (𝐴 = On → 𝐴 = On)
25 unieq 4580 . . . . . 6 (𝐴 = On → 𝐴 = On)
2624, 25eqeq12d 2785 . . . . 5 (𝐴 = On → (𝐴 = 𝐴 ↔ On = On))
27 eqeq1 2774 . . . . . . 7 (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
2827rexbidv 3199 . . . . . 6 (𝐴 = On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On On = suc 𝑥))
2928notbid 307 . . . . 5 (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
3026, 29bibi12d 334 . . . 4 (𝐴 = On → ((𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
3123, 30mpbiri 248 . . 3 (𝐴 = On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
3212, 31jaoi 837 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
331, 32sylbi 207 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 826   = wceq 1630  wcel 2144  wrex 3061  Vcvv 3349  c0 4061  ifcif 4223   cuni 4572  Ord word 5865  Oncon0 5866  suc csuc 5868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-tr 4885  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-suc 5872
This theorem is referenced by:  ordunisuc2  7190  ordzsl  7191  dflim3  7193  nnsuc  7228
  Copyright terms: Public domain W3C validator