Theorem ondif1i 43880

Description: Ordinal zero is less than every nonzero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8485. (Contributed by RP, 16-Jan-2025.)

Assertion

Ref	Expression
ondif1i	⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)

Proof of Theorem ondif1i

Step	Hyp	Ref	Expression
1		ondif1 8485	. 2 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
2	1	simprbi 502	1 ⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2149   ∖ cdif 3910  ∅c0 4294  Oncon0 6361  1_oc1o 8445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365  df-suc 6367  df-1o 8452

This theorem is referenced by: (None)