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Mirrors > Home > MPE Home > Th. List > unon | Structured version Visualization version GIF version |
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 4907 | . . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦) | |
2 | onelon 6388 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
3 | 2 | rexlimiva 3143 | . . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On) |
4 | 1, 3 | sylbi 216 | . . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On) |
5 | vex 3474 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | sucid 6445 | . . . 4 ⊢ 𝑥 ∈ suc 𝑥 |
7 | onsuc 7808 | . . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
8 | elunii 4908 | . . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On) | |
9 | 6, 7, 8 | sylancr 586 | . . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On) |
10 | 4, 9 | impbii 208 | . 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On) |
11 | 10 | eqriv 2725 | 1 ⊢ ∪ On = On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∃wrex 3066 ∪ cuni 4903 Oncon0 6363 suc csuc 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6366 df-on 6367 df-suc 6369 |
This theorem is referenced by: ordunisuc 7829 limon 7833 orduninsuc 7841 ordtoplem 35913 ordcmp 35925 onsupnmax 42650 |
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