Theorem onsucuni 2335

Description: A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41.

Assertion

Ref	Expression
onsucuni	⊢ (A ⊆ On → A ⊆ suc ∪A)

Proof of Theorem onsucuni

Step	Hyp	Ref	Expression
1		ssorduni 2249	. 2 ⊢ (A ⊆ On → Ord ∪A)
2		ssid 1519	. . 3 ⊢ ∪A ⊆ ∪A
3		ordunisssuc 2334	. . 3 ⊢ ((A ⊆ On ∧ Ord ∪A) → (∪A ⊆ ∪A ↔ A ⊆ suc ∪A))
4	2, 3	mpbii 168	. 2 ⊢ ((A ⊆ On ∧ Ord ∪A) → A ⊆ suc ∪A)
5	1, 4	mpdan 527	1 ⊢ (A ⊆ On → A ⊆ suc ∪A)

Syntax hints:   → wi 2   ∧ wa 196   ⊆ wss 1487  ∪cuni 1919  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  ordsucuni 2336  tz9.12lem3 3505

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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