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Mirrors > Home > MPE Home > Th. List > onzsl | Structured version Visualization version GIF version |
Description: An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
onzsl | ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3212 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
2 | eloni 5733 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordzsl 7045 | . . . 4 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) | |
4 | 3mix1 1230 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ∅) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
6 | 3mix2 1231 | . . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
8 | 3mix3 1232 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
9 | 5, 7, 8 | 3jaodan 1394 | . . . 4 ⊢ ((𝐴 ∈ V ∧ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
10 | 3, 9 | sylan2b 492 | . . 3 ⊢ ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
11 | 1, 2, 10 | syl2anc 693 | . 2 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
12 | 0elon 5778 | . . . 4 ⊢ ∅ ∈ On | |
13 | eleq1 2689 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On)) | |
14 | 12, 13 | mpbiri 248 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ On) |
15 | suceloni 7013 | . . . . 5 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
16 | eleq1 2689 | . . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On)) | |
17 | 15, 16 | syl5ibrcom 237 | . . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → 𝐴 ∈ On)) |
18 | 17 | rexlimiv 3027 | . . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 ∈ On) |
19 | limelon 5788 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On) | |
20 | 14, 18, 19 | 3jaoi 1391 | . 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On) |
21 | 11, 20 | impbii 199 | 1 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∅c0 3915 Ord word 5722 Oncon0 5723 Lim wlim 5724 suc csuc 5725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 |
This theorem is referenced by: oawordeulem 7634 r1pwss 8647 r1val1 8649 pwcfsdom 9405 winalim2 9518 rankcf 9599 dfrdg4 32058 |
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