Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3061 | . . . . . . 7 | |
2 | eloni 4267 | . . . . . . . 8 | |
3 | ordtr 4270 | . . . . . . . 8 | |
4 | 2, 3 | syl 14 | . . . . . . 7 |
5 | 1, 4 | syl6 33 | . . . . . 6 |
6 | 5 | ralrimiv 2481 | . . . . 5 |
7 | trint 4011 | . . . . 5 | |
8 | 6, 7 | syl 14 | . . . 4 |
9 | 8 | adantr 274 | . . 3 |
10 | nfv 1493 | . . . . 5 | |
11 | nfe1 1457 | . . . . 5 | |
12 | 10, 11 | nfan 1529 | . . . 4 |
13 | intssuni2m 3765 | . . . . . . . 8 | |
14 | unon 4397 | . . . . . . . 8 | |
15 | 13, 14 | sseqtrdi 3115 | . . . . . . 7 |
16 | 15 | sseld 3066 | . . . . . 6 |
17 | 16, 2 | syl6 33 | . . . . 5 |
18 | 17, 3 | syl6 33 | . . . 4 |
19 | 12, 18 | ralrimi 2480 | . . 3 |
20 | dford3 4259 | . . 3 | |
21 | 9, 19, 20 | sylanbrc 413 | . 2 |
22 | inteximm 4044 | . . . 4 | |
23 | 22 | adantl 275 | . . 3 |
24 | elong 4265 | . . 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 1453 wcel 1465 wral 2393 cvv 2660 wss 3041 cuni 3706 cint 3741 wtr 3996 word 4254 con0 4255 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 |
This theorem is referenced by: onintrab2im 4404 |
Copyright terms: Public domain | W3C validator |