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Mirrors > Home > MPE Home > Th. List > onun2i | Structured version Visualization version GIF version |
Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | ⊢ 𝐴 ∈ On |
on.2 | ⊢ 𝐵 ∈ On |
Ref | Expression |
---|---|
onun2i | ⊢ (𝐴 ∪ 𝐵) ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.1 | . 2 ⊢ 𝐴 ∈ On | |
2 | on.2 | . 2 ⊢ 𝐵 ∈ On | |
3 | onun2 6422 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ∪ 𝐵) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∪ cun 3906 Oncon0 6315 |
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 |
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 |
This theorem is referenced by: rankunb 9744 rankelun 9766 rankelpr 9767 inar1 10669 addsprop 34291 negsprop 34328 |
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