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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 3986 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o)) | |
2 | 1on 8534 | . . . . 5 ⊢ 1o ∈ On | |
3 | ontri1 6429 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴)) | |
4 | onsssuc 6485 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o)) | |
5 | df-2o 8523 | . . . . . . . 8 ⊢ 2o = suc 1o | |
6 | 5 | eleq2i 2836 | . . . . . . 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 690 | . . . 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 3973 ⊆ wss 3976 Oncon0 6395 suc csuc 6397 1oc1o 8515 2oc2o 8516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-tr 5284 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-ord 6398 df-on 6399 df-suc 6401 df-1o 8522 df-2o 8523 |
This theorem is referenced by: dif20el 8561 oeordi 8643 oewordi 8647 oaabs2 8705 omabs 8707 cnfcom3clem 9774 infxpenc2lem1 10088 onexoegt 43205 oege2 43269 rp-oelim2 43270 oeord2lim 43271 oeord2i 43272 oeord2com 43273 nnoeomeqom 43274 oenord1 43278 cantnftermord 43282 cantnfresb 43286 cantnf2 43287 omabs2 43294 omcl2 43295 |
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