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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 2931 | . 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
2 | ordeleqon 7782 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | unon 7832 | . . . . . . . . 9 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2734 | . . . . . . . 8 ⊢ On = ∪ On |
5 | id 22 | . . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On) | |
6 | unieq 4914 | . . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On) | |
7 | 4, 5, 6 | 3eqtr4a 2791 | . . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴) |
8 | 7 | orim2i 908 | . . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
9 | 2, 8 | sylbi 216 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
10 | 9 | orcomd 869 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On)) |
11 | 10 | ord 862 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On)) |
12 | orduniorsuc 7831 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) | |
13 | 12 | ord 862 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)) |
14 | onuni 7789 | . . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) | |
15 | ordtoplem.1 | . . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) | |
16 | eleq1a 2820 | . . . 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 | biimtrid 241 | 1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∪ cuni 4903 Ord word 6363 Oncon0 6364 suc csuc 6366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6367 df-on 6368 df-suc 6370 |
This theorem is referenced by: ordtop 35977 ordtopconn 35980 ordtopt0 35983 |
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