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Theorem onuninsuci 7543
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 6291 . . . . . 6 ¬ 𝐴𝐴
3 id 22 . . . . . . . 8 (𝐴 = 𝐴𝐴 = 𝐴)
4 df-suc 6191 . . . . . . . . . . . 12 suc 𝑥 = (𝑥 ∪ {𝑥})
54eqeq2i 2834 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝐴 = (𝑥 ∪ {𝑥}))
6 unieq 4840 . . . . . . . . . . 11 (𝐴 = (𝑥 ∪ {𝑥}) → 𝐴 = (𝑥 ∪ {𝑥}))
75, 6sylbi 218 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝐴 = (𝑥 ∪ {𝑥}))
8 uniun 4851 . . . . . . . . . . 11 (𝑥 ∪ {𝑥}) = ( 𝑥 {𝑥})
9 vex 3498 . . . . . . . . . . . . 13 𝑥 ∈ V
109unisn 4848 . . . . . . . . . . . 12 {𝑥} = 𝑥
1110uneq2i 4135 . . . . . . . . . . 11 ( 𝑥 {𝑥}) = ( 𝑥𝑥)
128, 11eqtri 2844 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = ( 𝑥𝑥)
137, 12syl6eq 2872 . . . . . . . . 9 (𝐴 = suc 𝑥 𝐴 = ( 𝑥𝑥))
14 tron 6208 . . . . . . . . . . . 12 Tr On
15 eleq1 2900 . . . . . . . . . . . . 13 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
161, 15mpbii 234 . . . . . . . . . . . 12 (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
17 trsuc 6269 . . . . . . . . . . . 12 ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
1814, 16, 17sylancr 587 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝑥 ∈ On)
19 eloni 6195 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
20 ordtr 6199 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
2119, 20syl 17 . . . . . . . . . . . 12 (𝑥 ∈ On → Tr 𝑥)
22 df-tr 5165 . . . . . . . . . . . 12 (Tr 𝑥 𝑥𝑥)
2321, 22sylib 219 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥𝑥)
2418, 23syl 17 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝑥𝑥)
25 ssequn1 4155 . . . . . . . . . 10 ( 𝑥𝑥 ↔ ( 𝑥𝑥) = 𝑥)
2624, 25sylib 219 . . . . . . . . 9 (𝐴 = suc 𝑥 → ( 𝑥𝑥) = 𝑥)
2713, 26eqtrd 2856 . . . . . . . 8 (𝐴 = suc 𝑥 𝐴 = 𝑥)
283, 27sylan9eqr 2878 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴 = 𝑥)
299sucid 6264 . . . . . . . . 9 𝑥 ∈ suc 𝑥
30 eleq2 2901 . . . . . . . . 9 (𝐴 = suc 𝑥 → (𝑥𝐴𝑥 ∈ suc 𝑥))
3129, 30mpbiri 259 . . . . . . . 8 (𝐴 = suc 𝑥𝑥𝐴)
3231adantr 481 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝑥𝐴)
3328, 32eqeltrd 2913 . . . . . 6 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴𝐴)
342, 33mto 198 . . . . 5 ¬ (𝐴 = suc 𝑥𝐴 = 𝐴)
3534imnani 401 . . . 4 (𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
3635rexlimivw 3282 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
37 onuni 7496 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
381, 37ax-mp 5 . . . 4 𝐴 ∈ On
391onuniorsuci 7542 . . . . 5 (𝐴 = 𝐴𝐴 = suc 𝐴)
4039ori 855 . . . 4 𝐴 = 𝐴𝐴 = suc 𝐴)
41 suceq 6250 . . . . 5 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
4241rspceeqv 3637 . . . 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 210 . 2 (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = 𝐴)
4544con2bii 359 1 (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3139  cun 3933  wss 3935  {csn 4559   cuni 4832  Tr wtr 5164  Ord word 6184  Oncon0 6185  suc csuc 6187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189  df-suc 6191
This theorem is referenced by:  orduninsuc  7546
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