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

Theorem orduninsuc 7784
Description: An ordinal class is equal to its union if and only if it is not the successor of an ordinal. Closed-form generalization of onuninsuci 7781. (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 7721 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 id 22 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3 unieq 4881 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
42, 3eqeq12d 2753 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅)))
5 eqeq1 2741 . . . . . . 7 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
65rexbidv 3176 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
76notbid 318 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
84, 7bibi12d 346 . . . 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 6376 . . . . . 6 ∅ ∈ On
109elimel 4560 . . . . 5 if(𝐴 ∈ On, 𝐴, ∅) ∈ On
1110onuninsuci 7781 . . . 4 (if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
128, 11dedth 4549 . . 3 (𝐴 ∈ On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13 unon 7771 . . . . . 6 On = On
1413eqcomi 2746 . . . . 5 On = On
15 onprc 7717 . . . . . . . 8 ¬ On ∈ V
16 vex 3452 . . . . . . . . . 10 𝑥 ∈ V
1716sucex 7746 . . . . . . . . 9 suc 𝑥 ∈ V
18 eleq1 2826 . . . . . . . . 9 (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
1917, 18mpbiri 258 . . . . . . . 8 (On = suc 𝑥 → On ∈ V)
2015, 19mto 196 . . . . . . 7 ¬ On = suc 𝑥
2120a1i 11 . . . . . 6 (𝑥 ∈ On → ¬ On = suc 𝑥)
2221nrex 3078 . . . . 5 ¬ ∃𝑥 ∈ On On = suc 𝑥
2314, 222th 264 . . . 4 (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24 id 22 . . . . . 6 (𝐴 = On → 𝐴 = On)
25 unieq 4881 . . . . . 6 (𝐴 = On → 𝐴 = On)
2624, 25eqeq12d 2753 . . . . 5 (𝐴 = On → (𝐴 = 𝐴 ↔ On = On))
27 eqeq1 2741 . . . . . . 7 (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
2827rexbidv 3176 . . . . . 6 (𝐴 = On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On On = suc 𝑥))
2928notbid 318 . . . . 5 (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
3026, 29bibi12d 346 . . . 4 (𝐴 = On → ((𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
3123, 30mpbiri 258 . . 3 (𝐴 = On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
3212, 31jaoi 856 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
331, 32sylbi 216 1 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 846   = wceq 1542  wcel 2107  wrex 3074  Vcvv 3448  c0 4287  ifcif 4491   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:  ordunisuc2  7785  ordzsl  7786  dflim3  7788  nnsuc  7825  onsupsucismax  41643
  Copyright terms: Public domain W3C validator