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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 4874 | . . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦) | |
2 | onelon 6347 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
3 | 2 | rexlimiva 3145 | . . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On) |
4 | 1, 3 | sylbi 216 | . . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On) |
5 | vex 3452 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | sucid 6404 | . . . 4 ⊢ 𝑥 ∈ suc 𝑥 |
7 | onsuc 7751 | . . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
8 | elunii 4875 | . . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On) | |
9 | 6, 7, 8 | sylancr 588 | . . 3 ⊢ (𝑥 ∈ On → 𝑥 ∈ ∪ On) |
10 | 4, 9 | impbii 208 | . 2 ⊢ (𝑥 ∈ ∪ On ↔ 𝑥 ∈ On) |
11 | 10 | eqriv 2734 | 1 ⊢ ∪ On = On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ∃wrex 3074 ∪ cuni 4870 Oncon0 6322 suc csuc 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-suc 6328 |
This theorem is referenced by: ordunisuc 7772 limon 7776 orduninsuc 7784 ordtoplem 34936 ordcmp 34948 onsupnmax 41591 |
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