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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 3459 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
2 | eloni 6169 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordzsl 7540 | . . . 4 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) | |
4 | 3mix1 1327 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
5 | 4 | adantl 485 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ∅) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
6 | 3mix2 1328 | . . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
7 | 6 | adantl 485 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
8 | 3mix3 1329 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
9 | 5, 7, 8 | 3jaodan 1427 | . . . 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 596 | . . 3 ⊢ ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
11 | 1, 2, 10 | syl2anc 587 | . 2 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
12 | 0elon 6212 | . . . 4 ⊢ ∅ ∈ On | |
13 | eleq1 2877 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On)) | |
14 | 12, 13 | mpbiri 261 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ On) |
15 | suceloni 7508 | . . . . 5 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
16 | eleq1 2877 | . . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On)) | |
17 | 15, 16 | syl5ibrcom 250 | . . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → 𝐴 ∈ On)) |
18 | 17 | rexlimiv 3239 | . . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 ∈ On) |
19 | limelon 6222 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On) | |
20 | 14, 18, 19 | 3jaoi 1424 | . 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On) |
21 | 11, 20 | impbii 212 | 1 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∅c0 4243 Ord word 6158 Oncon0 6159 Lim wlim 6160 suc csuc 6161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 |
This theorem is referenced by: oawordeulem 8163 r1pwss 9197 r1val1 9199 pwcfsdom 9994 winalim2 10107 rankcf 10188 dfrdg4 33525 |
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