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Mirrors > Home > MPE Home > Th. List > onuni | Structured version Visualization version GIF version |
Description: The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.) |
Ref | Expression |
---|---|
onuni | ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onss 7368 | . 2 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On) | |
2 | ssonuni 7364 | . 2 ⊢ (𝐴 ∈ On → (𝐴 ⊆ On → ∪ 𝐴 ∈ On)) | |
3 | 1, 2 | mpd 15 | 1 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2083 ⊆ wss 3865 ∪ cuni 4751 Oncon0 6073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-tr 5071 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-ord 6076 df-on 6077 |
This theorem is referenced by: onuninsuci 7418 oeeulem 8084 cnfcom3lem 9019 rankxpsuc 9164 dfac12lem2 9423 ttukeylem3 9786 r1limwun 10011 ontgval 33390 ordtoplem 33394 ordcmp 33406 1oequni2o 34201 rdgsucuni 34202 aomclem1 39160 |
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