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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. (Contributed by NM, 25-Nov-1995.) |
Ref | Expression |
---|---|
ord0eln0 | ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4235 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | ord0 6243 | . . . 4 ⊢ Ord ∅ | |
3 | noel 4231 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | ordtri2 6226 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 ∈ ∅ ↔ ¬ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) | |
5 | 4 | con2bid 358 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord ∅) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ ¬ 𝐴 ∈ ∅)) |
6 | 3, 5 | mpbiri 261 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
7 | 2, 6 | mpan2 691 | . . 3 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
8 | neor 3023 | . . 3 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) | |
9 | 7, 8 | sylib 221 | . 2 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) |
10 | 1, 9 | impbid2 229 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 Ord word 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 |
This theorem is referenced by: on0eln0 6246 dflim2 6247 0ellim 6253 0elsuc 7592 ordge1n0 8203 omwordi 8277 omass 8286 nnmord 8338 nnmwordi 8341 wemapwe 9290 elni2 10456 bnj529 32387 |
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