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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1228 | Structured version Visualization version Unicode version |
Description: Existence of a minimal element in certain classes: if is well-founded and set-like on , then every nonempty subclass of has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
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bnj1228.1 |
Ref | Expression |
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bnj1228 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj69 31078 | . 2 | |
2 | nfv 1843 | . . . 4 | |
3 | bnj1228.1 | . . . . . . 7 | |
4 | 3 | nfcii 2755 | . . . . . 6 |
5 | 4 | nfcri 2758 | . . . . 5 |
6 | nfv 1843 | . . . . . 6 | |
7 | 4, 6 | nfral 2945 | . . . . 5 |
8 | 5, 7 | nfan 1828 | . . . 4 |
9 | eleq1 2689 | . . . . 5 | |
10 | breq2 4657 | . . . . . . 7 | |
11 | 10 | notbid 308 | . . . . . 6 |
12 | 11 | ralbidv 2986 | . . . . 5 |
13 | 9, 12 | anbi12d 747 | . . . 4 |
14 | 2, 8, 13 | cbvex 2272 | . . 3 |
15 | df-rex 2918 | . . 3 | |
16 | df-rex 2918 | . . 3 | |
17 | 14, 15, 16 | 3bitr4i 292 | . 2 |
18 | 1, 17 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 w3a 1037 wal 1481 wex 1704 wcel 1990 wne 2794 wral 2912 wrex 2913 wss 3574 c0 3915 class class class wbr 4653 w-bnj15 30758 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-bnj17 30753 df-bnj14 30755 df-bnj13 30757 df-bnj15 30759 df-bnj18 30761 df-bnj19 30763 |
This theorem is referenced by: bnj1204 31080 bnj1311 31092 bnj1312 31126 |
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