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Theorem onsucuni 7517
 Description: A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.)
Assertion
Ref Expression
onsucuni (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)

Proof of Theorem onsucuni
StepHypRef Expression
1 ssorduni 7474 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 ssid 3964 . . 3 𝐴 𝐴
3 ordunisssuc 6265 . . 3 ((𝐴 ⊆ On ∧ Ord 𝐴) → ( 𝐴 𝐴𝐴 ⊆ suc 𝐴))
42, 3mpbii 235 . 2 ((𝐴 ⊆ On ∧ Ord 𝐴) → 𝐴 ⊆ suc 𝐴)
51, 4mpdan 685 1 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ⊆ wss 3909  ∪ cuni 4810  Ord word 6162  Oncon0 6163  suc csuc 6165 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 5175  ax-nul 5182  ax-pr 5302  ax-un 7435 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 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-tr 5145  df-eprel 5437  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-ord 6166  df-on 6167  df-suc 6169 This theorem is referenced by:  ordsucuni  7518  tz9.12lem3  9192  onssnum  9440  dfac12lem2  9544  ackbij1lem16  9631  cfslb2n  9664  hsmexlem1  9822
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