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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 4336 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | ord0 6428 | . . . 4 ⊢ Ord ∅ | |
3 | noel 4332 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | ordtri2 6410 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 ∈ ∅ ↔ ¬ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) | |
5 | 4 | con2bid 353 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord ∅) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ ¬ 𝐴 ∈ ∅)) |
6 | 3, 5 | mpbiri 257 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
7 | 2, 6 | mpan2 689 | . . 3 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
8 | neor 3023 | . . 3 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) | |
9 | 7, 8 | sylib 217 | . 2 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) |
10 | 1, 9 | impbid2 225 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∅c0 4324 Ord word 6374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-ord 6378 |
This theorem is referenced by: on0eln0 6431 dflim2 6432 0ellim 6438 0elsuc 7843 ordge1n0 8523 omwordi 8600 omass 8609 nnmord 8661 nnmwordi 8664 wemapwe 9736 elni2 10916 cuteq1 27855 bnj529 34554 ordeldif1o 42875 ordne0gt0 42876 dflim7 42888 |
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