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Theorem onuninsuci 7186
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 5977 . . . . . 6 ¬ 𝐴𝐴
3 id 22 . . . . . . . 8 (𝐴 = 𝐴𝐴 = 𝐴)
4 df-suc 5872 . . . . . . . . . . . 12 suc 𝑥 = (𝑥 ∪ {𝑥})
54eqeq2i 2782 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝐴 = (𝑥 ∪ {𝑥}))
6 unieq 4580 . . . . . . . . . . 11 (𝐴 = (𝑥 ∪ {𝑥}) → 𝐴 = (𝑥 ∪ {𝑥}))
75, 6sylbi 207 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝐴 = (𝑥 ∪ {𝑥}))
8 uniun 4591 . . . . . . . . . . 11 (𝑥 ∪ {𝑥}) = ( 𝑥 {𝑥})
9 vex 3352 . . . . . . . . . . . . 13 𝑥 ∈ V
109unisn 4587 . . . . . . . . . . . 12 {𝑥} = 𝑥
1110uneq2i 3913 . . . . . . . . . . 11 ( 𝑥 {𝑥}) = ( 𝑥𝑥)
128, 11eqtri 2792 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = ( 𝑥𝑥)
137, 12syl6eq 2820 . . . . . . . . 9 (𝐴 = suc 𝑥 𝐴 = ( 𝑥𝑥))
14 tron 5889 . . . . . . . . . . . 12 Tr On
15 eleq1 2837 . . . . . . . . . . . . 13 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
161, 15mpbii 223 . . . . . . . . . . . 12 (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
17 trsuc 5953 . . . . . . . . . . . 12 ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
1814, 16, 17sylancr 567 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝑥 ∈ On)
19 eloni 5876 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
20 ordtr 5880 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
2119, 20syl 17 . . . . . . . . . . . 12 (𝑥 ∈ On → Tr 𝑥)
22 df-tr 4885 . . . . . . . . . . . 12 (Tr 𝑥 𝑥𝑥)
2321, 22sylib 208 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥𝑥)
2418, 23syl 17 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝑥𝑥)
25 ssequn1 3932 . . . . . . . . . 10 ( 𝑥𝑥 ↔ ( 𝑥𝑥) = 𝑥)
2624, 25sylib 208 . . . . . . . . 9 (𝐴 = suc 𝑥 → ( 𝑥𝑥) = 𝑥)
2713, 26eqtrd 2804 . . . . . . . 8 (𝐴 = suc 𝑥 𝐴 = 𝑥)
283, 27sylan9eqr 2826 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴 = 𝑥)
299sucid 5947 . . . . . . . . 9 𝑥 ∈ suc 𝑥
30 eleq2 2838 . . . . . . . . 9 (𝐴 = suc 𝑥 → (𝑥𝐴𝑥 ∈ suc 𝑥))
3129, 30mpbiri 248 . . . . . . . 8 (𝐴 = suc 𝑥𝑥𝐴)
3231adantr 466 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝑥𝐴)
3328, 32eqeltrd 2849 . . . . . 6 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴𝐴)
342, 33mto 188 . . . . 5 ¬ (𝐴 = suc 𝑥𝐴 = 𝐴)
3534imnani 387 . . . 4 (𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
3635rexlimivw 3176 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
37 onuni 7139 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
381, 37ax-mp 5 . . . 4 𝐴 ∈ On
391onuniorsuci 7185 . . . . 5 (𝐴 = 𝐴𝐴 = suc 𝐴)
4039ori 841 . . . 4 𝐴 = 𝐴𝐴 = suc 𝐴)
41 suceq 5933 . . . . . 6 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
4241eqeq2d 2780 . . . . 5 (𝑥 = 𝐴 → (𝐴 = suc 𝑥𝐴 = suc 𝐴))
4342rspcev 3458 . . . 4 (( 𝐴 ∈ On ∧ 𝐴 = suc 𝐴) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4438, 40, 43sylancr 567 . . 3 𝐴 = 𝐴 → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4536, 44impbii 199 . 2 (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = 𝐴)
4645con2bii 346 1 (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382   = wceq 1630  wcel 2144  wrex 3061  cun 3719  wss 3721  {csn 4314   cuni 4572  Tr wtr 4884  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-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:  orduninsuc  7189
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