Theorem ondif1i 42604

Description: Ordinal zero is less than every non-zero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8513. (Contributed by RP, 16-Jan-2025.)

Assertion

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

Proof of Theorem ondif1i

Step	Hyp	Ref	Expression
1		ondif1 8513	. 2 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
2	1	simprbi 496	1 ⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2099   ∖ cdif 3941  ∅c0 4318  Oncon0 6363  1_oc1o 8471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-suc 6369  df-1o 8478

This theorem is referenced by: (None)