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Mirrors > Home > NFE Home > Th. List > abvor0 | Unicode version |
Description: The class builder of a wff not containing the abstraction variable is either the universal class or the empty set. (Contributed by Mario Carneiro, 29-Aug-2013.) |
Ref | Expression |
---|---|
abvor0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . . 6 | |
2 | vex 2862 | . . . . . . 7 | |
3 | 2 | a1i 10 | . . . . . 6 |
4 | 1, 3 | 2thd 231 | . . . . 5 |
5 | 4 | abbi1dv 2469 | . . . 4 |
6 | 5 | con3i 127 | . . 3 |
7 | id 19 | . . . . 5 | |
8 | noel 3554 | . . . . . 6 | |
9 | 8 | a1i 10 | . . . . 5 |
10 | 7, 9 | 2falsed 340 | . . . 4 |
11 | 10 | abbi1dv 2469 | . . 3 |
12 | 6, 11 | syl 15 | . 2 |
13 | 12 | orri 365 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wo 357 wceq 1642 wcel 1710 cab 2339 cvv 2859 c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 |
This theorem is referenced by: abexv 4324 |
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