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Theorem nlimsucg 4226
Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
nlimsucg (A 𝑉 → ¬ Lim suc A)

Proof of Theorem nlimsucg
StepHypRef Expression
1 limord 4081 . . . . . 6 (Lim suc A → Ord suc A)
2 ordsuc 4225 . . . . . 6 (Ord A ↔ Ord suc A)
31, 2sylibr 137 . . . . 5 (Lim suc A → Ord A)
4 limuni 4082 . . . . 5 (Lim suc A → suc A = suc A)
53, 4jca 290 . . . 4 (Lim suc A → (Ord A suc A = suc A))
6 ordtr 4064 . . . . . . . 8 (Ord A → Tr A)
7 unisucg 4100 . . . . . . . . 9 (A 𝑉 → (Tr A suc A = A))
87biimpa 280 . . . . . . . 8 ((A 𝑉 Tr A) → suc A = A)
96, 8sylan2 270 . . . . . . 7 ((A 𝑉 Ord A) → suc A = A)
109eqeq2d 2033 . . . . . 6 ((A 𝑉 Ord A) → (suc A = suc A ↔ suc A = A))
11 ordirr 4209 . . . . . . . . 9 (Ord A → ¬ A A)
12 eleq2 2083 . . . . . . . . . 10 (suc A = A → (A suc AA A))
1312notbid 579 . . . . . . . . 9 (suc A = A → (¬ A suc A ↔ ¬ A A))
1411, 13syl5ibrcom 146 . . . . . . . 8 (Ord A → (suc A = A → ¬ A suc A))
15 sucidg 4102 . . . . . . . . 9 (A 𝑉A suc A)
1615con3i 549 . . . . . . . 8 A suc A → ¬ A 𝑉)
1714, 16syl6 29 . . . . . . 7 (Ord A → (suc A = A → ¬ A 𝑉))
1817adantl 262 . . . . . 6 ((A 𝑉 Ord A) → (suc A = A → ¬ A 𝑉))
1910, 18sylbid 139 . . . . 5 ((A 𝑉 Ord A) → (suc A = suc A → ¬ A 𝑉))
2019expimpd 345 . . . 4 (A 𝑉 → ((Ord A suc A = suc A) → ¬ A 𝑉))
215, 20syl5 28 . . 3 (A 𝑉 → (Lim suc A → ¬ A 𝑉))
2221con2d 542 . 2 (A 𝑉 → (A 𝑉 → ¬ Lim suc A))
2322pm2.43i 43 1 (A 𝑉 → ¬ Lim suc A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1228   wcel 1374   cuni 3554  Tr wtr 3828  Ord word 4048  Lim wlim 4050  suc csuc 4051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-ilim 4055  df-suc 4057
This theorem is referenced by: (None)
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