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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 3957 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o)) | |
2 | 1on 8499 | . . . . 5 ⊢ 1o ∈ On | |
3 | ontri1 6403 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴)) | |
4 | onsssuc 6459 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o)) | |
5 | df-2o 8488 | . . . . . . . 8 ⊢ 2o = suc 1o | |
6 | 5 | eleq2i 2821 | . . . . . . 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 205 ∧ wa 395 ∈ wcel 2099 ∖ cdif 3944 ⊆ wss 3947 Oncon0 6369 suc csuc 6371 1oc1o 8480 2oc2o 8481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6372 df-on 6373 df-suc 6375 df-1o 8487 df-2o 8488 |
This theorem is referenced by: dif20el 8526 oeordi 8608 oewordi 8612 oaabs2 8670 omabs 8672 cnfcom3clem 9729 infxpenc2lem1 10043 onexoegt 42672 oege2 42736 rp-oelim2 42737 oeord2lim 42738 oeord2i 42739 oeord2com 42740 nnoeomeqom 42741 oenord1 42745 cantnftermord 42749 cantnfresb 42753 cantnf2 42754 omabs2 42761 omcl2 42762 |
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