| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > onpsssuc | Structured version Visualization version GIF version | ||
| Description: An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| Ref | Expression |
|---|---|
| onpsssuc | ⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sucidg 6465 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴) | |
| 2 | eloni 6394 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 3 | ordsuc 7833 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
| 4 | 2, 3 | sylib 218 | . . 3 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
| 5 | ordelpss 6412 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) | |
| 6 | 2, 4, 5 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴)) |
| 7 | 1, 6 | mpbid 232 | 1 ⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 ⊊ wpss 3952 Ord word 6383 Oncon0 6384 suc csuc 6386 |
| 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 |
| This theorem is referenced by: ackbij1lem15 10273 |
| Copyright terms: Public domain | W3C validator |