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Theorem onpsssuc 4323
Description: An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
onpsssuc (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)

Proof of Theorem onpsssuc
StepHypRef Expression
1 elirr 4294 . . . 4 ¬ 𝐴𝐴
2 sucidg 4181 . . . . 5 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
3 eleq2 2117 . . . . 5 (𝐴 = suc 𝐴 → (𝐴𝐴𝐴 ∈ suc 𝐴))
42, 3syl5ibrcom 150 . . . 4 (𝐴 ∈ On → (𝐴 = suc 𝐴𝐴𝐴))
51, 4mtoi 600 . . 3 (𝐴 ∈ On → ¬ 𝐴 = suc 𝐴)
6 sssucid 4180 . . 3 𝐴 ⊆ suc 𝐴
75, 6jctil 299 . 2 (𝐴 ∈ On → (𝐴 ⊆ suc 𝐴 ∧ ¬ 𝐴 = suc 𝐴))
8 dfpss2 3057 . 2 (𝐴 ⊊ suc 𝐴 ↔ (𝐴 ⊆ suc 𝐴 ∧ ¬ 𝐴 = suc 𝐴))
97, 8sylibr 141 1 (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101   = wceq 1259  wcel 1409  wss 2945  wpss 2946  Oncon0 4128  suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pss 2961  df-sn 3409  df-suc 4136
This theorem is referenced by: (None)
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