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

Theorem onuninsuci 7687
Description: A limit ordinal is not a successor ordinal. (Contributed by NM, 18-Feb-2004.)
Hypothesis
Ref Expression
onssi.1 𝐴 ∈ On
Assertion
Ref Expression
onuninsuci (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem onuninsuci
StepHypRef Expression
1 onssi.1 . . . . . . 7 𝐴 ∈ On
21onirri 6373 . . . . . 6 ¬ 𝐴𝐴
3 id 22 . . . . . . . 8 (𝐴 = 𝐴𝐴 = 𝐴)
4 df-suc 6272 . . . . . . . . . . . 12 suc 𝑥 = (𝑥 ∪ {𝑥})
54eqeq2i 2751 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝐴 = (𝑥 ∪ {𝑥}))
6 unieq 4850 . . . . . . . . . . 11 (𝐴 = (𝑥 ∪ {𝑥}) → 𝐴 = (𝑥 ∪ {𝑥}))
75, 6sylbi 216 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝐴 = (𝑥 ∪ {𝑥}))
8 uniun 4864 . . . . . . . . . . 11 (𝑥 ∪ {𝑥}) = ( 𝑥 {𝑥})
9 vex 3436 . . . . . . . . . . . . 13 𝑥 ∈ V
109unisn 4861 . . . . . . . . . . . 12 {𝑥} = 𝑥
1110uneq2i 4094 . . . . . . . . . . 11 ( 𝑥 {𝑥}) = ( 𝑥𝑥)
128, 11eqtri 2766 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = ( 𝑥𝑥)
137, 12eqtrdi 2794 . . . . . . . . 9 (𝐴 = suc 𝑥 𝐴 = ( 𝑥𝑥))
14 tron 6289 . . . . . . . . . . . 12 Tr On
15 eleq1 2826 . . . . . . . . . . . . 13 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
161, 15mpbii 232 . . . . . . . . . . . 12 (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
17 trsuc 6350 . . . . . . . . . . . 12 ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
1814, 16, 17sylancr 587 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝑥 ∈ On)
19 eloni 6276 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
20 ordtr 6280 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
2119, 20syl 17 . . . . . . . . . . . 12 (𝑥 ∈ On → Tr 𝑥)
22 df-tr 5192 . . . . . . . . . . . 12 (Tr 𝑥 𝑥𝑥)
2321, 22sylib 217 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥𝑥)
2418, 23syl 17 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝑥𝑥)
25 ssequn1 4114 . . . . . . . . . 10 ( 𝑥𝑥 ↔ ( 𝑥𝑥) = 𝑥)
2624, 25sylib 217 . . . . . . . . 9 (𝐴 = suc 𝑥 → ( 𝑥𝑥) = 𝑥)
2713, 26eqtrd 2778 . . . . . . . 8 (𝐴 = suc 𝑥 𝐴 = 𝑥)
283, 27sylan9eqr 2800 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴 = 𝑥)
299sucid 6345 . . . . . . . . 9 𝑥 ∈ suc 𝑥
30 eleq2 2827 . . . . . . . . 9 (𝐴 = suc 𝑥 → (𝑥𝐴𝑥 ∈ suc 𝑥))
3129, 30mpbiri 257 . . . . . . . 8 (𝐴 = suc 𝑥𝑥𝐴)
3231adantr 481 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝑥𝐴)
3328, 32eqeltrd 2839 . . . . . 6 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴𝐴)
342, 33mto 196 . . . . 5 ¬ (𝐴 = suc 𝑥𝐴 = 𝐴)
3534imnani 401 . . . 4 (𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
3635rexlimivw 3211 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
37 onuni 7638 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
381, 37ax-mp 5 . . . 4 𝐴 ∈ On
391onuniorsuci 7686 . . . . 5 (𝐴 = 𝐴𝐴 = suc 𝐴)
4039ori 858 . . . 4 𝐴 = 𝐴𝐴 = suc 𝐴)
41 suceq 6331 . . . . 5 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
4241rspceeqv 3575 . . . 4 (( 𝐴 ∈ On ∧ 𝐴 = suc 𝐴) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4338, 40, 42sylancr 587 . . 3 𝐴 = 𝐴 → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4436, 43impbii 208 . 2 (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = 𝐴)
4544con2bii 358 1 (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  cun 3885  wss 3887  {csn 4561   cuni 4839  Tr wtr 5191  Ord word 6265  Oncon0 6266  suc csuc 6268
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-suc 6272
This theorem is referenced by:  orduninsuc  7690
  Copyright terms: Public domain W3C validator