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| Mirrors > Home > MPE Home > Th. List > ordge1n0 | Structured version Visualization version GIF version | ||
| Description: An ordinal greater than or equal to 1 is nonzero. (Contributed by NM, 21-Dec-2004.) | 
| Ref | Expression | 
|---|---|
| ordge1n0 | ⊢ (Ord 𝐴 → (1o ⊆ 𝐴 ↔ 𝐴 ≠ ∅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ordgt0ge1 8531 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o ⊆ 𝐴)) | |
| 2 | ord0eln0 6439 | . 2 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
| 3 | 1, 2 | bitr3d 281 | 1 ⊢ (Ord 𝐴 → (1o ⊆ 𝐴 ↔ 𝐴 ≠ ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 Ord word 6383 1oc1o 8499 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 df-1o 8506 | 
| This theorem is referenced by: om00 8613 bday1s 27876 finxpsuc 37399 oege1 43319 | 
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