Theorem ordsucuni 7780

Description: An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)

Assertion

Ref	Expression
ordsucuni	⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴)

Proof of Theorem ordsucuni

Step	Hyp	Ref	Expression
1		ordsson 7737	. 2 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
2		onsucuni 7779	. 2 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
3	1, 2	syl 17	1 ⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3890  ∪ cuni 4851  Ord word 6323  Oncon0 6324  suc csuc 6326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6327  df-on 6328  df-suc 6330

This theorem is referenced by:  orduniorsuc  7781