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Mirrors > Home > MPE Home > Th. List > onun2i | Structured version Visualization version Unicode version |
Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) |
Ref | Expression |
---|---|
on.1 | |
on.2 |
Ref | Expression |
---|---|
onun2i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | on.2 | . . . 4 | |
2 | 1 | onordi 5832 | . . 3 |
3 | on.1 | . . . 4 | |
4 | 3 | onordi 5832 | . . 3 |
5 | ordtri2or 5822 | . . 3 | |
6 | 2, 4, 5 | mp2an 708 | . 2 |
7 | 3 | oneluni 5840 | . . . 4 |
8 | 7, 3 | syl6eqel 2709 | . . 3 |
9 | ssequn1 3783 | . . . 4 | |
10 | eleq1 2689 | . . . . 5 | |
11 | 1, 10 | mpbiri 248 | . . . 4 |
12 | 9, 11 | sylbi 207 | . . 3 |
13 | 8, 12 | jaoi 394 | . 2 |
14 | 6, 13 | ax-mp 5 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wo 383 wceq 1483 wcel 1990 cun 3572 wss 3574 word 5722 con0 5723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 |
This theorem is referenced by: rankunb 8713 rankelun 8735 rankelpr 8736 inar1 9597 |
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