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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 3972 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o)) | |
2 | 1on 8516 | . . . . 5 ⊢ 1o ∈ On | |
3 | ontri1 6419 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴)) | |
4 | onsssuc 6475 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o)) | |
5 | df-2o 8505 | . . . . . . . 8 ⊢ 2o = suc 1o | |
6 | 5 | eleq2i 2830 | . . . . . . 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 2105 ∖ cdif 3959 ⊆ wss 3962 Oncon0 6385 suc csuc 6387 1oc1o 8497 2oc2o 8498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 df-suc 6391 df-1o 8504 df-2o 8505 |
This theorem is referenced by: dif20el 8541 oeordi 8623 oewordi 8627 oaabs2 8685 omabs 8687 cnfcom3clem 9742 infxpenc2lem1 10056 onexoegt 43232 oege2 43296 rp-oelim2 43297 oeord2lim 43298 oeord2i 43299 oeord2com 43300 nnoeomeqom 43301 oenord1 43305 cantnftermord 43309 cantnfresb 43313 cantnf2 43314 omabs2 43321 omcl2 43322 |
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