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Theorem ordsssuc 6253
Description: An ordinal is a subset of another ordinal if and only if it belongs to its successor. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordsssuc ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ∈ suc 𝐵))

Proof of Theorem ordsssuc
StepHypRef Expression
1 eloni 6177 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordsseleq 6196 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
31, 2sylan 582 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
4 elsucg 6234 . . 3 (𝐴 ∈ On → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
54adantr 483 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
63, 5bitr4d 284 1 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ∈ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1537  wcel 2114  wss 3913  Ord word 6166  Oncon0 6167  suc csuc 6169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-ord 6170  df-on 6171  df-suc 6173
This theorem is referenced by:  onsssuc  6254  ordunisssuc  6269  ordpwsuc  7508  ordsucun  7518  cantnflt  9113  cantnflem1  9130  noetalem3  33227
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