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Mirrors > Home > MPE Home > Th. List > axnulALT | Structured version Visualization version Unicode version |
Description: Alternate proof of axnul 4525, proved directly from ax-rep 4508 using none of the equality axioms ax-7 1859 through ax-c14 32527 provided we accept sp 1957 as an axiom. Replace sp 1957 with the obsolete ax-c5 32519 to see this in 'show traceback'. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axnulALT |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rep 4508 |
. . 3
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2 | sp 1957 |
. . . . . 6
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3 | 2 | con2i 124 |
. . . . 5
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4 | df-ex 1672 |
. . . . 5
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5 | 3, 4 | sylibr 217 |
. . . 4
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6 | fal 1459 |
. . . . . 6
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7 | sp 1957 |
. . . . . 6
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8 | 6, 7 | mto 181 |
. . . . 5
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9 | 8 | pm2.21i 136 |
. . . 4
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10 | 5, 9 | mpg 1679 |
. . 3
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11 | 1, 10 | mpg 1679 |
. 2
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12 | 8 | intnan 928 |
. . . . . 6
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13 | 12 | nex 1686 |
. . . . 5
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14 | 13 | nbn 354 |
. . . 4
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15 | 14 | albii 1699 |
. . 3
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16 | 15 | exbii 1726 |
. 2
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17 | 11, 16 | mpbir 214 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-12 1950 ax-rep 4508 |
This theorem depends on definitions: df-bi 190 df-an 378 df-tru 1455 df-fal 1458 df-ex 1672 |
This theorem is referenced by: (None) |
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