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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 2941 | . 2 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴) | |
| 2 | ordeleqon 7802 | . . . . . 6 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
| 3 | unon 7851 | . . . . . . . . 9 ⊢ ∪ On = On | |
| 4 | 3 | eqcomi 2746 | . . . . . . . 8 ⊢ On = ∪ On |
| 5 | id 22 | . . . . . . . 8 ⊢ (𝐴 = On → 𝐴 = On) | |
| 6 | unieq 4918 | . . . . . . . 8 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On) | |
| 7 | 4, 5, 6 | 3eqtr4a 2803 | . . . . . . 7 ⊢ (𝐴 = On → 𝐴 = ∪ 𝐴) |
| 8 | 7 | orim2i 911 | . . . . . 6 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
| 9 | 2, 8 | sylbi 217 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = ∪ 𝐴)) |
| 10 | 9 | orcomd 872 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 ∈ On)) |
| 11 | 10 | ord 865 | . . 3 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 ∈ On)) |
| 12 | orduniorsuc 7850 | . . . 4 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)) | |
| 13 | 12 | ord 865 | . . 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 2836 | . . . 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 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∪ cuni 4907 Ord word 6383 Oncon0 6384 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: ordtop 36437 ordtopconn 36440 ordtopt0 36443 |
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