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Theorem orduninsuc 7834
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 7831. (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 7771 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 id 22 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3 unieq 4918 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
42, 3eqeq12d 2746 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅)))
5 eqeq1 2734 . . . . . . 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 317 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
84, 7bibi12d 344 . . . 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 6417 . . . . . 6 ∅ ∈ On
109elimel 4596 . . . . 5 if(𝐴 ∈ On, 𝐴, ∅) ∈ On
1110onuninsuci 7831 . . . 4 (if(𝐴 ∈ On, 𝐴, ∅) = if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
128, 11dedth 4585 . . 3 (𝐴 ∈ On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13 unon 7821 . . . . . 6 On = On
1413eqcomi 2739 . . . . 5 On = On
15 onprc 7767 . . . . . . . 8 ¬ On ∈ V
16 vex 3476 . . . . . . . . . 10 𝑥 ∈ V
1716sucex 7796 . . . . . . . . 9 suc 𝑥 ∈ V
18 eleq1 2819 . . . . . . . . 9 (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
1917, 18mpbiri 257 . . . . . . . 8 (On = suc 𝑥 → On ∈ V)
2015, 19mto 196 . . . . . . 7 ¬ On = suc 𝑥
2120a1i 11 . . . . . 6 (𝑥 ∈ On → ¬ On = suc 𝑥)
2221nrex 3072 . . . . 5 ¬ ∃𝑥 ∈ On On = suc 𝑥
2314, 222th 263 . . . 4 (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24 id 22 . . . . . 6 (𝐴 = On → 𝐴 = On)
25 unieq 4918 . . . . . 6 (𝐴 = On → 𝐴 = On)
2624, 25eqeq12d 2746 . . . . 5 (𝐴 = On → (𝐴 = 𝐴 ↔ On = On))
27 eqeq1 2734 . . . . . . 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 317 . . . . 5 (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
3026, 29bibi12d 344 . . . 4 (𝐴 = On → ((𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
3123, 30mpbiri 257 . . 3 (𝐴 = On → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
3212, 31jaoi 853 . 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 843   = wceq 1539  wcel 2104  wrex 3068  Vcvv 3472  c0 4321  ifcif 4527   cuni 4907  Ord word 6362  Oncon0 6363  suc csuc 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367  df-suc 6369
This theorem is referenced by:  ordunisuc2  7835  ordzsl  7836  dflim3  7838  nnsuc  7875  onsupsucismax  42331
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