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Mirrors > Home > MPE Home > Th. List > ord0eln0 | Structured version Visualization version GIF version |
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 4347 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | ord0 6439 | . . . 4 ⊢ Ord ∅ | |
3 | noel 4344 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | ordtri2 6421 | . . . . . 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 3032 | . . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 Ord word 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 |
This theorem is referenced by: on0eln0 6442 dflim2 6443 0ellim 6449 0elsuc 7855 ordge1n0 8531 omwordi 8608 omass 8617 nnmord 8669 nnmwordi 8672 wemapwe 9735 elni2 10915 cuteq1 27893 bnj529 34734 ordeldif1o 43250 ordne0gt0 43251 dflim7 43263 |
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