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Theorem onuninsuci 7238
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 6014 . . . . . 6 ¬ 𝐴𝐴
3 id 22 . . . . . . . 8 (𝐴 = 𝐴𝐴 = 𝐴)
4 df-suc 5914 . . . . . . . . . . . 12 suc 𝑥 = (𝑥 ∪ {𝑥})
54eqeq2i 2777 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝐴 = (𝑥 ∪ {𝑥}))
6 unieq 4602 . . . . . . . . . . 11 (𝐴 = (𝑥 ∪ {𝑥}) → 𝐴 = (𝑥 ∪ {𝑥}))
75, 6sylbi 208 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝐴 = (𝑥 ∪ {𝑥}))
8 uniun 4615 . . . . . . . . . . 11 (𝑥 ∪ {𝑥}) = ( 𝑥 {𝑥})
9 vex 3353 . . . . . . . . . . . . 13 𝑥 ∈ V
109unisn 4610 . . . . . . . . . . . 12 {𝑥} = 𝑥
1110uneq2i 3926 . . . . . . . . . . 11 ( 𝑥 {𝑥}) = ( 𝑥𝑥)
128, 11eqtri 2787 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = ( 𝑥𝑥)
137, 12syl6eq 2815 . . . . . . . . 9 (𝐴 = suc 𝑥 𝐴 = ( 𝑥𝑥))
14 tron 5931 . . . . . . . . . . . 12 Tr On
15 eleq1 2832 . . . . . . . . . . . . 13 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
161, 15mpbii 224 . . . . . . . . . . . 12 (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
17 trsuc 5992 . . . . . . . . . . . 12 ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
1814, 16, 17sylancr 581 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝑥 ∈ On)
19 eloni 5918 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
20 ordtr 5922 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
2119, 20syl 17 . . . . . . . . . . . 12 (𝑥 ∈ On → Tr 𝑥)
22 df-tr 4912 . . . . . . . . . . . 12 (Tr 𝑥 𝑥𝑥)
2321, 22sylib 209 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥𝑥)
2418, 23syl 17 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝑥𝑥)
25 ssequn1 3945 . . . . . . . . . 10 ( 𝑥𝑥 ↔ ( 𝑥𝑥) = 𝑥)
2624, 25sylib 209 . . . . . . . . 9 (𝐴 = suc 𝑥 → ( 𝑥𝑥) = 𝑥)
2713, 26eqtrd 2799 . . . . . . . 8 (𝐴 = suc 𝑥 𝐴 = 𝑥)
283, 27sylan9eqr 2821 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴 = 𝑥)
299sucid 5987 . . . . . . . . 9 𝑥 ∈ suc 𝑥
30 eleq2 2833 . . . . . . . . 9 (𝐴 = suc 𝑥 → (𝑥𝐴𝑥 ∈ suc 𝑥))
3129, 30mpbiri 249 . . . . . . . 8 (𝐴 = suc 𝑥𝑥𝐴)
3231adantr 472 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝑥𝐴)
3328, 32eqeltrd 2844 . . . . . 6 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴𝐴)
342, 33mto 188 . . . . 5 ¬ (𝐴 = suc 𝑥𝐴 = 𝐴)
3534imnani 389 . . . 4 (𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
3635rexlimivw 3176 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
37 onuni 7191 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
381, 37ax-mp 5 . . . 4 𝐴 ∈ On
391onuniorsuci 7237 . . . . 5 (𝐴 = 𝐴𝐴 = suc 𝐴)
4039ori 887 . . . 4 𝐴 = 𝐴𝐴 = suc 𝐴)
41 suceq 5973 . . . . 5 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
4241rspceeqv 3479 . . . 4 (( 𝐴 ∈ On ∧ 𝐴 = suc 𝐴) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4338, 40, 42sylancr 581 . . 3 𝐴 = 𝐴 → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4436, 43impbii 200 . 2 (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = 𝐴)
4544con2bii 348 1 (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384   = wceq 1652  wcel 2155  wrex 3056  cun 3730  wss 3732  {csn 4334   cuni 4594  Tr wtr 4911  Ord word 5907  Oncon0 5908  suc csuc 5910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-suc 5914
This theorem is referenced by:  orduninsuc  7241
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