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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 4867 | . . . 4 ⊢ (𝑥 ∈ ∪ On ↔ ∃𝑦 ∈ On 𝑥 ∈ 𝑦) | |
2 | onelon 6340 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ On) | |
3 | 2 | rexlimiva 3142 | . . . 4 ⊢ (∃𝑦 ∈ On 𝑥 ∈ 𝑦 → 𝑥 ∈ On) |
4 | 1, 3 | sylbi 216 | . . 3 ⊢ (𝑥 ∈ ∪ On → 𝑥 ∈ On) |
5 | vex 3447 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | 5 | sucid 6397 | . . . 4 ⊢ 𝑥 ∈ suc 𝑥 |
7 | onsuc 7738 | . . . 4 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
8 | elunii 4868 | . . . 4 ⊢ ((𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ On) → 𝑥 ∈ ∪ On) | |
9 | 6, 7, 8 | sylancr 587 | . . 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 1541 ∈ wcel 2106 ∃wrex 3071 ∪ cuni 4863 Oncon0 6315 suc csuc 6317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-tr 5221 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6318 df-on 6319 df-suc 6321 |
This theorem is referenced by: ordunisuc 7759 limon 7763 orduninsuc 7771 ordtoplem 34838 ordcmp 34850 onsupnmax 41464 |
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