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| Description: A nonempty ordinal contains the empty set. Lemma 1.10 of [Schloeder] p. 2. (Contributed by NM, 25-Nov-1995.) | 
| Ref | Expression | 
|---|---|
| ord0eln0 | ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ne0i 4340 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
| 2 | ord0 6436 | . . . 4 ⊢ Ord ∅ | |
| 3 | noel 4337 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
| 4 | ordtri2 6418 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 ∈ ∅ ↔ ¬ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) | |
| 5 | 4 | con2bid 354 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord ∅) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ ¬ 𝐴 ∈ ∅)) | 
| 6 | 3, 5 | mpbiri 258 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | 
| 7 | 2, 6 | mpan2 691 | . . 3 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) | 
| 8 | neor 3033 | . . 3 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) | |
| 9 | 7, 8 | sylib 218 | . 2 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) | 
| 10 | 1, 9 | impbid2 226 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∅c0 4332 Ord word 6382 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 | 
| This theorem is referenced by: on0eln0 6439 dflim2 6440 0ellim 6446 0elsuc 7856 ordge1n0 8533 omwordi 8610 omass 8619 nnmord 8671 nnmwordi 8674 wemapwe 9738 elni2 10918 cuteq1 27879 bnj529 34756 ordeldif1o 43278 ordne0gt0 43279 dflim7 43291 | 
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