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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 2999 | . 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
2 | ordeleqon 7248 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | unon 7291 | . . . . . . . . 9 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2833 | . . . . . . . 8 ⊢ On = ∪ On |
5 | id 22 | . . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On) | |
6 | unieq 4665 | . . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On) | |
7 | 4, 5, 6 | 3eqtr4a 2886 | . . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴) |
8 | 7 | orim2i 941 | . . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
9 | 2, 8 | sylbi 209 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
10 | 9 | orcomd 904 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On)) |
11 | 10 | ord 897 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On)) |
12 | orduniorsuc 7290 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) | |
13 | 12 | ord 897 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)) |
14 | onuni 7253 | . . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) | |
15 | ordtoplem.1 | . . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) | |
16 | eleq1a 2900 | . . . 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 234 | 1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 ∪ cuni 4657 Ord word 5961 Oncon0 5962 suc csuc 5964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-tr 4975 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-ord 5965 df-on 5966 df-suc 5968 |
This theorem is referenced by: ordtop 32967 ordtopconn 32970 ordtopt0 32973 |
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