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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 4186 | . 2 ⊢ (∅ ∈ 𝐴 → 𝐴 ≠ ∅) | |
2 | ord0 6081 | . . . 4 ⊢ Ord ∅ | |
3 | noel 4183 | . . . . 5 ⊢ ¬ 𝐴 ∈ ∅ | |
4 | ordtri2 6064 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 ∈ ∅ ↔ ¬ (𝐴 = ∅ ∨ ∅ ∈ 𝐴))) | |
5 | 4 | con2bid 347 | . . . . 5 ⊢ ((Ord 𝐴 ∧ Ord ∅) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ ¬ 𝐴 ∈ ∅)) |
6 | 3, 5 | mpbiri 250 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord ∅) → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
7 | 2, 6 | mpan2 678 | . . 3 ⊢ (Ord 𝐴 → (𝐴 = ∅ ∨ ∅ ∈ 𝐴)) |
8 | neor 3059 | . . 3 ⊢ ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ↔ (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) | |
9 | 7, 8 | sylib 210 | . 2 ⊢ (Ord 𝐴 → (𝐴 ≠ ∅ → ∅ ∈ 𝐴)) |
10 | 1, 9 | impbid2 218 | 1 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∅c0 4178 Ord word 6028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-tr 5031 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-ord 6032 |
This theorem is referenced by: on0eln0 6084 dflim2 6085 0ellim 6091 0elsuc 7366 ordge1n0 7925 omwordi 7998 omass 8007 nnmord 8059 nnmwordi 8062 wemapwe 8954 elni2 10097 bnj529 31666 |
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