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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 3924 | . 2 ⊢ (𝐴 ∈ (On ∖ 2o) ↔ (𝐴 ∈ On ∧ ¬ 𝐴 ∈ 2o)) | |
2 | 1on 8428 | . . . . 5 ⊢ 1o ∈ On | |
3 | ontri1 6355 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ ¬ 1o ∈ 𝐴)) | |
4 | onsssuc 6411 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 1o ∈ On) → (𝐴 ⊆ 1o ↔ 𝐴 ∈ suc 1o)) | |
5 | df-2o 8417 | . . . . . . . 8 ⊢ 2o = suc 1o | |
6 | 5 | eleq2i 2826 | . . . . . . 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 356 | . . 3 ⊢ (𝐴 ∈ On → (¬ 𝐴 ∈ 2o ↔ 1o ∈ 𝐴)) |
11 | 10 | pm5.32i 576 | . 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 397 ∈ wcel 2107 ∖ cdif 3911 ⊆ wss 3914 Oncon0 6321 suc csuc 6323 1oc1o 8409 2oc2o 8410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-tr 5227 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-ord 6324 df-on 6325 df-suc 6327 df-1o 8416 df-2o 8417 |
This theorem is referenced by: dif20el 8455 oeordi 8538 oewordi 8542 oaabs2 8599 omabs 8601 cnfcom3clem 9649 infxpenc2lem1 9963 onexoegt 41625 oege2 41689 rp-oelim2 41690 oeord2lim 41691 oeord2i 41692 oeord2com 41693 nnoeomeqom 41694 oenord1 41698 cantnftermord 41702 cantnfresb 41706 cantnf2 41707 omabs2 41714 omcl2 41715 |
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