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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. (Contributed by Mario Carneiro, 10-Oct-2014.) |
Ref | Expression |
---|---|
axext.1 |
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axext.2 |
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Ref | Expression |
---|---|
axext |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axext.1 |
. . . . . 6
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2 | wv 64 |
. . . . . 6
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3 | 1, 2 | wc 50 |
. . . . 5
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4 | 3 | wl 66 |
. . . 4
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5 | axext.2 |
. . . . . . . 8
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6 | 5, 2 | wc 50 |
. . . . . . 7
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7 | 3, 6 | weqi 76 |
. . . . . 6
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8 | 7 | ax4 150 |
. . . . 5
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9 | wal 134 |
. . . . . 6
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10 | 7 | wl 66 |
. . . . . 6
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11 | wv 64 |
. . . . . 6
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12 | 9, 11 | ax-17 105 |
. . . . . 6
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13 | 7, 11 | ax-hbl1 103 |
. . . . . 6
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14 | 9, 10, 11, 12, 13 | hbc 110 |
. . . . 5
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15 | 3, 8, 14 | leqf 181 |
. . . 4
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16 | 15 | ax-cb1 29 |
. . . . 5
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17 | 1 | eta 178 |
. . . . 5
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18 | 16, 17 | a1i 28 |
. . . 4
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19 | 5 | eta 178 |
. . . . 5
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20 | 16, 19 | a1i 28 |
. . . 4
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21 | 4, 15, 18, 20 | 3eqtr3i 97 |
. . 3
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22 | wtru 43 |
. . 3
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23 | 21, 22 | adantl 56 |
. 2
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24 | 23 | ex 158 |
1
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Colors of variables: type var term |
Syntax hints: tv 1
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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-wctl 31 ax-wctr 32 ax-weq 40 ax-refl 42 ax-eqmp 45 ax-ded 46 ax-wct 47 ax-wc 49 ax-ceq 51 ax-wv 63 ax-wl 65 ax-beta 67 ax-distrc 68 ax-leq 69 ax-distrl 70 ax-wov 71 ax-eqtypi 77 ax-eqtypri 80 ax-hbl1 103 ax-17 105 ax-inst 113 ax-eta 177 |
This theorem depends on definitions: df-ov 73 df-al 126 df-an 128 df-im 129 |
This theorem is referenced by: (None) |
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