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Mirrors > Home > ILE Home > Th. List > tfri1dALT | Unicode version |
Description: Alternate proof of tfri1d 6200 in terms of tfr1on 6215.
Although this does show that the tfr1on 6215 proof is general enough to also prove tfri1d 6200, the tfri1d 6200 proof is simpler in places because it does not need to deal with being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.) |
Ref | Expression |
---|---|
tfri1dALT.1 | recs |
tfri1dALT.2 |
Ref | Expression |
---|---|
tfri1dALT |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrfun 6185 | . . . 4 recs | |
2 | tfri1dALT.1 | . . . . 5 recs | |
3 | 2 | funeqi 5114 | . . . 4 recs |
4 | 1, 3 | mpbir 145 | . . 3 |
5 | 4 | a1i 9 | . 2 |
6 | eqid 2117 | . . . . . 6 | |
7 | 6 | tfrlem8 6183 | . . . . 5 recs |
8 | 2 | dmeqi 4710 | . . . . . 6 recs |
9 | ordeq 4264 | . . . . . 6 recs recs | |
10 | 8, 9 | ax-mp 5 | . . . . 5 recs |
11 | 7, 10 | mpbir 145 | . . . 4 |
12 | ordsson 4378 | . . . 4 | |
13 | 11, 12 | mp1i 10 | . . 3 |
14 | tfri1dALT.2 | . . . . . . . . . 10 | |
15 | simpl 108 | . . . . . . . . . . 11 | |
16 | 15 | alimi 1416 | . . . . . . . . . 10 |
17 | 14, 16 | syl 14 | . . . . . . . . 9 |
18 | 17 | 19.21bi 1522 | . . . . . . . 8 |
19 | 18 | adantr 274 | . . . . . . 7 |
20 | ordon 4372 | . . . . . . . 8 | |
21 | 20 | a1i 9 | . . . . . . 7 |
22 | simpr 109 | . . . . . . . . . . 11 | |
23 | 22 | alimi 1416 | . . . . . . . . . 10 |
24 | fveq2 5389 | . . . . . . . . . . . 12 | |
25 | 24 | eleq1d 2186 | . . . . . . . . . . 11 |
26 | 25 | spv 1816 | . . . . . . . . . 10 |
27 | 14, 23, 26 | 3syl 17 | . . . . . . . . 9 |
28 | 27 | adantr 274 | . . . . . . . 8 |
29 | 28 | 3ad2ant1 987 | . . . . . . 7 |
30 | suceloni 4387 | . . . . . . . . 9 | |
31 | unon 4397 | . . . . . . . . 9 | |
32 | 30, 31 | eleq2s 2212 | . . . . . . . 8 |
33 | 32 | adantl 275 | . . . . . . 7 |
34 | suceloni 4387 | . . . . . . . 8 | |
35 | 34 | adantl 275 | . . . . . . 7 |
36 | 2, 19, 21, 29, 33, 35 | tfr1on 6215 | . . . . . 6 |
37 | vex 2663 | . . . . . . 7 | |
38 | 37 | sucid 4309 | . . . . . 6 |
39 | ssel2 3062 | . . . . . 6 | |
40 | 36, 38, 39 | sylancl 409 | . . . . 5 |
41 | 40 | ex 114 | . . . 4 |
42 | 41 | ssrdv 3073 | . . 3 |
43 | 13, 42 | eqssd 3084 | . 2 |
44 | df-fn 5096 | . 2 | |
45 | 5, 43, 44 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1314 wceq 1316 wcel 1465 cab 2103 wral 2393 wrex 2394 cvv 2660 wss 3041 cuni 3706 word 4254 con0 4255 csuc 4257 cdm 4509 cres 4511 wfun 5087 wfn 5088 cfv 5093 recscrecs 6169 |
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-in1 588 ax-in2 589 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-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-recs 6170 |
This theorem is referenced by: (None) |
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