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| Mirrors > Home > MPE Home > Th. List > ondif2 | Structured version Visualization version GIF version | ||
| Description: Two ways to say that 𝐴 is an ordinal greater than one. (Contributed by Mario Carneiro, 21-May-2015.) |
| Ref | Expression |
|---|---|
| ondif2 | ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldif 3961 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o)) | |
| 2 | 1on 8518 | . . . . 5 ⊢ 1o ∈ On | |
| 3 | ontri1 6418 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴)) | |
| 4 | onsssuc 6474 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o)) | |
| 5 | df-2o 8507 | . . . . . . . 8 ⊢ 2o = suc 1o | |
| 6 | 5 | eleq2i 2833 | . . . . . . 7 ⊢ (𝐴 ∈ 2o ↔ 𝐴 ∈ suc 1o) |
| 7 | 4, 6 | bitr4di 289 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ 2o)) |
| 8 | 3, 7 | bitr3d 281 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o)) |
| 9 | 2, 8 | mpan2 691 | . . . 4 ⊢ (𝐴 ∈ On → (¬ 1o ∈ 𝐴 ↔ 𝐴 ∈ 2o)) |
| 10 | 9 | con1bid 355 | . . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o ∈ 𝐴)) |
| 11 | 10 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) |
| 12 | 1, 11 | bitri 275 | 1 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ 1o ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 Oncon0 6384 suc csuc 6386 1oc1o 8499 2oc2o 8500 |
| 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 df-2o 8507 |
| This theorem is referenced by: dif20el 8543 oeordi 8625 oewordi 8629 oaabs2 8687 omabs 8689 cnfcom3clem 9745 infxpenc2lem1 10059 onexoegt 43256 oege2 43320 rp-oelim2 43321 oeord2lim 43322 oeord2i 43323 oeord2com 43324 nnoeomeqom 43325 oenord1 43329 cantnftermord 43333 cantnfresb 43337 cantnf2 43338 omabs2 43345 omcl2 43346 |
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