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Theorem onuninsuci 7547
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 6290 . . . . . 6 ¬ 𝐴𝐴
3 id 22 . . . . . . . 8 (𝐴 = 𝐴𝐴 = 𝐴)
4 df-suc 6190 . . . . . . . . . . . 12 suc 𝑥 = (𝑥 ∪ {𝑥})
54eqeq2i 2832 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝐴 = (𝑥 ∪ {𝑥}))
6 unieq 4838 . . . . . . . . . . 11 (𝐴 = (𝑥 ∪ {𝑥}) → 𝐴 = (𝑥 ∪ {𝑥}))
75, 6sylbi 219 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝐴 = (𝑥 ∪ {𝑥}))
8 uniun 4849 . . . . . . . . . . 11 (𝑥 ∪ {𝑥}) = ( 𝑥 {𝑥})
9 vex 3496 . . . . . . . . . . . . 13 𝑥 ∈ V
109unisn 4846 . . . . . . . . . . . 12 {𝑥} = 𝑥
1110uneq2i 4134 . . . . . . . . . . 11 ( 𝑥 {𝑥}) = ( 𝑥𝑥)
128, 11eqtri 2842 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = ( 𝑥𝑥)
137, 12syl6eq 2870 . . . . . . . . 9 (𝐴 = suc 𝑥 𝐴 = ( 𝑥𝑥))
14 tron 6207 . . . . . . . . . . . 12 Tr On
15 eleq1 2898 . . . . . . . . . . . . 13 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
161, 15mpbii 235 . . . . . . . . . . . 12 (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
17 trsuc 6268 . . . . . . . . . . . 12 ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
1814, 16, 17sylancr 589 . . . . . . . . . . 11 (𝐴 = suc 𝑥𝑥 ∈ On)
19 eloni 6194 . . . . . . . . . . . . 13 (𝑥 ∈ On → Ord 𝑥)
20 ordtr 6198 . . . . . . . . . . . . 13 (Ord 𝑥 → Tr 𝑥)
2119, 20syl 17 . . . . . . . . . . . 12 (𝑥 ∈ On → Tr 𝑥)
22 df-tr 5164 . . . . . . . . . . . 12 (Tr 𝑥 𝑥𝑥)
2321, 22sylib 220 . . . . . . . . . . 11 (𝑥 ∈ On → 𝑥𝑥)
2418, 23syl 17 . . . . . . . . . 10 (𝐴 = suc 𝑥 𝑥𝑥)
25 ssequn1 4154 . . . . . . . . . 10 ( 𝑥𝑥 ↔ ( 𝑥𝑥) = 𝑥)
2624, 25sylib 220 . . . . . . . . 9 (𝐴 = suc 𝑥 → ( 𝑥𝑥) = 𝑥)
2713, 26eqtrd 2854 . . . . . . . 8 (𝐴 = suc 𝑥 𝐴 = 𝑥)
283, 27sylan9eqr 2876 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴 = 𝑥)
299sucid 6263 . . . . . . . . 9 𝑥 ∈ suc 𝑥
30 eleq2 2899 . . . . . . . . 9 (𝐴 = suc 𝑥 → (𝑥𝐴𝑥 ∈ suc 𝑥))
3129, 30mpbiri 260 . . . . . . . 8 (𝐴 = suc 𝑥𝑥𝐴)
3231adantr 483 . . . . . . 7 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝑥𝐴)
3328, 32eqeltrd 2911 . . . . . 6 ((𝐴 = suc 𝑥𝐴 = 𝐴) → 𝐴𝐴)
342, 33mto 199 . . . . 5 ¬ (𝐴 = suc 𝑥𝐴 = 𝐴)
3534imnani 403 . . . 4 (𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
3635rexlimivw 3280 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = 𝐴)
37 onuni 7500 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ On)
381, 37ax-mp 5 . . . 4 𝐴 ∈ On
391onuniorsuci 7546 . . . . 5 (𝐴 = 𝐴𝐴 = suc 𝐴)
4039ori 857 . . . 4 𝐴 = 𝐴𝐴 = suc 𝐴)
41 suceq 6249 . . . . 5 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
4241rspceeqv 3636 . . . 4 (( 𝐴 ∈ On ∧ 𝐴 = suc 𝐴) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4338, 40, 42sylancr 589 . . 3 𝐴 = 𝐴 → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
4436, 43impbii 211 . 2 (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = 𝐴)
4544con2bii 360 1 (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208  wa 398   = wceq 1530  wcel 2107  wrex 3137  cun 3932  wss 3934  {csn 4559   cuni 4830  Tr wtr 5163  Ord word 6183  Oncon0 6184  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190
This theorem is referenced by:  orduninsuc  7550
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