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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordtoplem | Structured version Visualization version GIF version |
Description: Membership of the class of successor ordinals. (Contributed by Chen-Pang He, 1-Nov-2015.) |
Ref | Expression |
---|---|
ordtoplem.1 | ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) |
Ref | Expression |
---|---|
ordtoplem | ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 3017 | . 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
2 | ordeleqon 7497 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | unon 7540 | . . . . . . . . 9 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2830 | . . . . . . . 8 ⊢ On = ∪ On |
5 | id 22 | . . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On) | |
6 | unieq 4840 | . . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On) | |
7 | 4, 5, 6 | 3eqtr4a 2882 | . . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴) |
8 | 7 | orim2i 907 | . . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
9 | 2, 8 | sylbi 219 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
10 | 9 | orcomd 867 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On)) |
11 | 10 | ord 860 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On)) |
12 | orduniorsuc 7539 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) | |
13 | 12 | ord 860 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)) |
14 | onuni 7502 | . . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) | |
15 | ordtoplem.1 | . . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) | |
16 | eleq1a 2908 | . . . 4 ⊢ (suc ∪ 𝐴 ∈ 𝑆 → (𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆)) | |
17 | 14, 15, 16 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 = suc ∪ 𝐴 → 𝐴 ∈ 𝑆)) |
18 | 11, 13, 17 | syl6c 70 | . 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ 𝑆)) |
19 | 1, 18 | syl5bi 244 | 1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∪ cuni 4832 Ord word 6185 Oncon0 6186 suc csuc 6188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-ord 6189 df-on 6190 df-suc 6192 |
This theorem is referenced by: ordtop 33779 ordtopconn 33782 ordtopt0 33785 |
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