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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 2939 | . 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
2 | ordeleqon 7801 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | unon 7851 | . . . . . . . . 9 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2744 | . . . . . . . 8 ⊢ On = ∪ On |
5 | id 22 | . . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On) | |
6 | unieq 4923 | . . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On) | |
7 | 4, 5, 6 | 3eqtr4a 2801 | . . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴) |
8 | 7 | orim2i 910 | . . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
9 | 2, 8 | sylbi 217 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
10 | 9 | orcomd 871 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On)) |
11 | 10 | ord 864 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On)) |
12 | orduniorsuc 7850 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) | |
13 | 12 | ord 864 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)) |
14 | onuni 7808 | . . . 4 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) | |
15 | ordtoplem.1 | . . . 4 ⊢ (∪ 𝐴 ∈ On → suc ∪ 𝐴 ∈ 𝑆) | |
16 | eleq1a 2834 | . . . 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 242 | 1 ⊢ (Ord 𝐴 → (𝐴 ≠ ∪ 𝐴 → 𝐴 ∈ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∪ cuni 4912 Ord word 6385 Oncon0 6386 suc csuc 6388 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-suc 6392 |
This theorem is referenced by: ordtop 36419 ordtopconn 36422 ordtopt0 36425 |
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