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Mirrors > Home > MPE Home > Th. List > Mathboxes > wsuclem | Structured version Visualization version Unicode version |
Description: Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
Ref | Expression |
---|---|
wsuclem.1 | |
wsuclem.2 | Se |
wsuclem.3 | |
wsuclem.4 |
Ref | Expression |
---|---|
wsuclem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wsuclem.1 | . . 3 | |
2 | wsuclem.2 | . . 3 Se | |
3 | predss 5687 | . . . 4 | |
4 | 3 | a1i 11 | . . 3 |
5 | wsuclem.3 | . . . . 5 | |
6 | dfpred3g 5691 | . . . . 5 | |
7 | 5, 6 | syl 17 | . . . 4 |
8 | 5 | elexd 3214 | . . . . 5 |
9 | wsuclem.4 | . . . . 5 | |
10 | rabn0 3958 | . . . . . . 7 | |
11 | brcnvg 5303 | . . . . . . . . 9 | |
12 | 11 | ancoms 469 | . . . . . . . 8 |
13 | 12 | rexbidva 3049 | . . . . . . 7 |
14 | 10, 13 | syl5bb 272 | . . . . . 6 |
15 | 14 | biimpar 502 | . . . . 5 |
16 | 8, 9, 15 | syl2anc 693 | . . . 4 |
17 | 7, 16 | eqnetrd 2861 | . . 3 |
18 | tz6.26 5711 | . . 3 Se | |
19 | 1, 2, 4, 17, 18 | syl22anc 1327 | . 2 |
20 | dfpred3g 5691 | . . . . 5 | |
21 | 5, 20 | syl 17 | . . . 4 |
22 | 21 | rexeqdv 3145 | . . 3 |
23 | breq1 4656 | . . . . 5 | |
24 | 23 | rexrab 3370 | . . . 4 |
25 | noel 3919 | . . . . . . . . . . . 12 | |
26 | simp2r 1088 | . . . . . . . . . . . . 13 | |
27 | 26 | eleq2d 2687 | . . . . . . . . . . . 12 |
28 | 25, 27 | mtbiri 317 | . . . . . . . . . . 11 |
29 | vex 3203 | . . . . . . . . . . . . 13 | |
30 | 29 | a1i 11 | . . . . . . . . . . . 12 |
31 | simp3 1063 | . . . . . . . . . . . 12 | |
32 | elpredg 5694 | . . . . . . . . . . . 12 | |
33 | 30, 31, 32 | syl2anc 693 | . . . . . . . . . . 11 |
34 | 28, 33 | mtbid 314 | . . . . . . . . . 10 |
35 | 34 | 3expa 1265 | . . . . . . . . 9 |
36 | 35 | ralrimiva 2966 | . . . . . . . 8 |
37 | 36 | expr 643 | . . . . . . 7 |
38 | simp1rl 1126 | . . . . . . . . . . . 12 | |
39 | simp1rr 1127 | . . . . . . . . . . . 12 | |
40 | 5 | adantr 481 | . . . . . . . . . . . . . 14 |
41 | 40 | 3ad2ant1 1082 | . . . . . . . . . . . . 13 |
42 | 29 | elpred 5693 | . . . . . . . . . . . . 13 |
43 | 41, 42 | syl 17 | . . . . . . . . . . . 12 |
44 | 38, 39, 43 | mpbir2and 957 | . . . . . . . . . . 11 |
45 | simp3 1063 | . . . . . . . . . . 11 | |
46 | breq1 4656 | . . . . . . . . . . . 12 | |
47 | 46 | rspcev 3309 | . . . . . . . . . . 11 |
48 | 44, 45, 47 | syl2anc 693 | . . . . . . . . . 10 |
49 | 48 | 3expia 1267 | . . . . . . . . 9 |
50 | 49 | ralrimiva 2966 | . . . . . . . 8 |
51 | 50 | expr 643 | . . . . . . 7 |
52 | 37, 51 | anim12d 586 | . . . . . 6 |
53 | 52 | ancomsd 470 | . . . . 5 |
54 | 53 | reximdva 3017 | . . . 4 |
55 | 24, 54 | syl5bi 232 | . . 3 |
56 | 22, 55 | sylbid 230 | . 2 |
57 | 19, 56 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wral 2912 wrex 2913 crab 2916 cvv 3200 wss 3574 c0 3915 class class class wbr 4653 Se wse 5071 wwe 5072 ccnv 5113 cpred 5679 |
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-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 |
This theorem is referenced by: wsuccl 31776 wsuclb 31777 |
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