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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 3400 | . . 3 ⊢ (𝐴 ∈ On → 𝐴 ∈ V) | |
2 | eloni 5951 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | ordzsl 7279 | . . . 4 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) | |
4 | 3mix1 1430 | . . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
5 | 4 | adantl 474 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ∅) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
6 | 3mix2 1431 | . . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
7 | 6 | adantl 474 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
8 | 3mix3 1432 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) | |
9 | 5, 7, 8 | 3jaodan 1556 | . . . 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 588 | . . 3 ⊢ ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
11 | 1, 2, 10 | syl2anc 580 | . 2 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
12 | 0elon 5994 | . . . 4 ⊢ ∅ ∈ On | |
13 | eleq1 2866 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On)) | |
14 | 12, 13 | mpbiri 250 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ On) |
15 | suceloni 7247 | . . . . 5 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
16 | eleq1 2866 | . . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On)) | |
17 | 15, 16 | syl5ibrcom 239 | . . . 4 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → 𝐴 ∈ On)) |
18 | 17 | rexlimiv 3208 | . . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → 𝐴 ∈ On) |
19 | limelon 6004 | . . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On) | |
20 | 14, 18, 19 | 3jaoi 1553 | . 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On) |
21 | 11, 20 | impbii 201 | 1 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 ∨ w3o 1107 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 Vcvv 3385 ∅c0 4115 Ord word 5940 Oncon0 5941 Lim wlim 5942 suc csuc 5943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-tr 4946 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 |
This theorem is referenced by: oawordeulem 7874 r1pwss 8897 r1val1 8899 pwcfsdom 9693 winalim2 9806 rankcf 9887 dfrdg4 32571 |
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