Theorem ondif1i 41945

Description: Ordinal zero is less than every non-zero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8496. (Contributed by RP, 16-Jan-2025.)

Assertion

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

Proof of Theorem ondif1i

Step	Hyp	Ref	Expression
1		ondif1 8496	. 2 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
2	1	simprbi 498	1 ⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2107   ∖ cdif 3944  ∅c0 4321  Oncon0 6361  1_oc1o 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367  df-1o 8461

This theorem is referenced by: (None)