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Mirrors > Home > ILE Home > Th. List > onintonm | Unicode version |
Description: The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
onintonm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3131 | . . . . . . 7 | |
2 | eloni 4347 | . . . . . . . 8 | |
3 | ordtr 4350 | . . . . . . . 8 | |
4 | 2, 3 | syl 14 | . . . . . . 7 |
5 | 1, 4 | syl6 33 | . . . . . 6 |
6 | 5 | ralrimiv 2536 | . . . . 5 |
7 | trint 4089 | . . . . 5 | |
8 | 6, 7 | syl 14 | . . . 4 |
9 | 8 | adantr 274 | . . 3 |
10 | nfv 1515 | . . . . 5 | |
11 | nfe1 1483 | . . . . 5 | |
12 | 10, 11 | nfan 1552 | . . . 4 |
13 | intssuni2m 3842 | . . . . . . . 8 | |
14 | unon 4482 | . . . . . . . 8 | |
15 | 13, 14 | sseqtrdi 3185 | . . . . . . 7 |
16 | 15 | sseld 3136 | . . . . . 6 |
17 | 16, 2 | syl6 33 | . . . . 5 |
18 | 17, 3 | syl6 33 | . . . 4 |
19 | 12, 18 | ralrimi 2535 | . . 3 |
20 | dford3 4339 | . . 3 | |
21 | 9, 19, 20 | sylanbrc 414 | . 2 |
22 | inteximm 4122 | . . . 4 | |
23 | 22 | adantl 275 | . . 3 |
24 | elong 4345 | . . 3 | |
25 | 23, 24 | syl 14 | . 2 |
26 | 21, 25 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wex 1479 wcel 2135 wral 2442 cvv 2721 wss 3111 cuni 3783 cint 3818 wtr 4074 word 4334 con0 4335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 |
This theorem is referenced by: onintrab2im 4489 |
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