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Mirrors > Home > HOLE Home > Th. List > axext | Unicode version |
Description: Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It states that two sets are identical if they contain the same elements. Axiom Ext of [BellMachover] p. 461. |
Ref | Expression |
---|---|
axext.1 | |
axext.2 |
Ref | Expression |
---|---|
axext |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axext.1 | . . . . . 6 | |
2 | wv 58 | . . . . . 6 | |
3 | 1, 2 | wc 45 | . . . . 5 |
4 | 3 | wl 59 | . . . 4 |
5 | axext.2 | . . . . . . . 8 | |
6 | 5, 2 | wc 45 | . . . . . . 7 |
7 | 3, 6 | weqi 68 | . . . . . 6 |
8 | 7 | ax4 140 | . . . . 5 |
9 | wal 124 | . . . . . 6 | |
10 | 7 | wl 59 | . . . . . 6 |
11 | wv 58 | . . . . . 6 | |
12 | 9, 11 | ax-17 95 | . . . . . 6 |
13 | 7, 11 | ax-hbl1 93 | . . . . . 6 |
14 | 9, 10, 11, 12, 13 | hbc 100 | . . . . 5 |
15 | 3, 8, 14 | leqf 169 | . . . 4 |
16 | 15 | ax-cb1 29 | . . . . 5 |
17 | 1 | eta 166 | . . . . 5 |
18 | 16, 17 | a1i 28 | . . . 4 |
19 | 5 | eta 166 | . . . . 5 |
20 | 16, 19 | a1i 28 | . . . 4 |
21 | 4, 15, 18, 20 | 3eqtr3i 87 | . . 3 |
22 | wtru 40 | . . 3 | |
23 | 21, 22 | adantl 51 | . 2 |
24 | 23 | ex 148 | 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 |
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 ax-eta 165 |
This theorem depends on definitions: df-ov 65 df-al 116 df-an 118 df-im 119 |
This theorem is referenced by: (None) |
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