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Mirrors > Home > HOLE Home > Th. List > axpow | Unicode version |
Description: Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. |
Ref | Expression |
---|---|
axpow.1 |
Ref | Expression |
---|---|
axpow |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wtru 40 | . . . . 5 | |
2 | wal 124 | . . . . . 6 | |
3 | wim 127 | . . . . . . . 8 | |
4 | wv 58 | . . . . . . . . 9 | |
5 | wv 58 | . . . . . . . . 9 | |
6 | 4, 5 | wc 45 | . . . . . . . 8 |
7 | axpow.1 | . . . . . . . . 9 | |
8 | 7, 5 | wc 45 | . . . . . . . 8 |
9 | 3, 6, 8 | wov 64 | . . . . . . 7 |
10 | 9 | wl 59 | . . . . . 6 |
11 | 2, 10 | wc 45 | . . . . 5 |
12 | 1, 11 | simpl 22 | . . . 4 |
13 | 12 | ex 148 | . . 3 |
14 | 13 | alrimiv 141 | . 2 |
15 | wal 124 | . . . 4 | |
16 | wv 58 | . . . . . . 7 | |
17 | 16, 4 | wc 45 | . . . . . 6 |
18 | 3, 11, 17 | wov 64 | . . . . 5 |
19 | 18 | wl 59 | . . . 4 |
20 | 15, 19 | wc 45 | . . 3 |
21 | 1 | wl 59 | . . 3 |
22 | 16, 21 | weqi 68 | . . . . . . . . 9 |
23 | 22 | id 25 | . . . . . . . 8 |
24 | 16, 4, 23 | ceq1 79 | . . . . . . 7 |
25 | wv 58 | . . . . . . . . . . 11 | |
26 | 25, 4 | weqi 68 | . . . . . . . . . 10 |
27 | 26, 1 | eqid 73 | . . . . . . . . 9 |
28 | 1, 4, 27 | cl 106 | . . . . . . . 8 |
29 | 22, 28 | a1i 28 | . . . . . . 7 |
30 | 17, 24, 29 | eqtri 85 | . . . . . 6 |
31 | 3, 11, 17, 30 | oveq2 91 | . . . . 5 |
32 | 18, 31 | leq 81 | . . . 4 |
33 | 15, 19, 32 | ceq2 80 | . . 3 |
34 | 20, 21, 33 | cla4ev 159 | . 2 |
35 | 14, 34 | syl 16 | 1 |
Colors of variables: type var term |
Syntax hints: tv 1 ht 2 hb 3 kc 5 kl 6 ke 7 kt 8 kbr 9 wffMMJ2 11 wffMMJ2t 12 tim 111 tal 112 tex 113 |
This theorem was proved from axioms: ax-syl 15 ax-jca 17 ax-simpl 20 ax-simpr 21 ax-id 24 ax-trud 26 ax-cb1 29 ax-cb2 30 ax-refl 39 ax-eqmp 42 ax-ded 43 ax-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-distrl 63 ax-hbl1 93 ax-17 95 ax-inst 103 |
This theorem depends on definitions: df-ov 65 df-al 116 df-an 118 df-im 119 df-ex 121 |
This theorem is referenced by: (None) |
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