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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fun2dmnopgexmpl | Structured version Visualization version Unicode version |
Description: A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) |
Ref | Expression |
---|---|
fun2dmnopgexmpl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ne1 10684 |
. . . . . . . 8
![]() ![]() ![]() ![]() | |
2 | 1 | neii 2628 |
. . . . . . 7
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3 | 2 | intnanr 927 |
. . . . . 6
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4 | 3 | intnanr 927 |
. . . . 5
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5 | 4 | gen2 1672 |
. . . 4
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6 | eqeq1 2457 |
. . . . . . . 8
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7 | c0ex 9642 |
. . . . . . . . 9
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8 | 1ex 9643 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
9 | vex 3050 |
. . . . . . . . 9
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10 | vex 3050 |
. . . . . . . . 9
![]() ![]() ![]() ![]() | |
11 | 7, 8, 8, 8, 9, 10 | propeqop 39009 |
. . . . . . . 8
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12 | 6, 11 | syl6bb 265 |
. . . . . . 7
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13 | 12 | notbid 296 |
. . . . . 6
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14 | 13 | albidv 1769 |
. . . . 5
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15 | 14 | albidv 1769 |
. . . 4
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16 | 5, 15 | mpbiri 237 |
. . 3
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17 | 2nexaln 1704 |
. . 3
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18 | 16, 17 | sylibr 216 |
. 2
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19 | elvv 4896 |
. 2
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20 | 18, 19 | sylnibr 307 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pr 4642 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-mulcl 9606 ax-i2m1 9612 ax-1ne0 9613 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-v 3049 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-sn 3971 df-pr 3973 df-op 3977 df-opab 4465 df-xp 4843 |
This theorem is referenced by: (None) |
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