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| Description: The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.) | 
| Ref | Expression | 
|---|---|
| unon | ⊢ ∪ On = On | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eluni2 4910 | . . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦) | |
| 2 | onelon 6408 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
| 3 | 2 | rexlimiva 3146 | . . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On) | 
| 4 | 1, 3 | sylbi 217 | . . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On) | 
| 5 | vex 3483 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 6 | 5 | sucid 6465 | . . . 4 ⊢ 𝑥 ∈ suc 𝑥 | 
| 7 | onsuc 7832 | . . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
| 8 | elunii 4911 | . . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On) | |
| 9 | 6, 7, 8 | sylancr 587 | . . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On) | 
| 10 | 4, 9 | impbii 209 | . 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On) | 
| 11 | 10 | eqriv 2733 | 1 ⊢ ∪ On = On | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ∪ cuni 4906 Oncon0 6383 suc csuc 6385 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 df-suc 6389 | 
| This theorem is referenced by: ordunisuc 7853 limon 7857 orduninsuc 7865 ordtoplem 36437 ordcmp 36449 onsupnmax 43245 | 
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