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Mirrors > Home > MPE Home > Th. List > zfcndac | Structured version Visualization version Unicode version |
Description: Axiom of Choice ax-ac 9281, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
zfcndac |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axacnd 9434 | . . 3 | |
2 | 19.3v 1897 | . . . . . 6 | |
3 | 2 | imbi1i 339 | . . . . 5 |
4 | 3 | 2albii 1748 | . . . 4 |
5 | 4 | exbii 1774 | . . 3 |
6 | 1, 5 | mpbi 220 | . 2 |
7 | equequ2 1953 | . . . . . . . . . 10 | |
8 | 7 | bibi2d 332 | . . . . . . . . 9 |
9 | elequ2 2004 | . . . . . . . . . . . . 13 | |
10 | 9 | anbi2d 740 | . . . . . . . . . . . 12 |
11 | elequ2 2004 | . . . . . . . . . . . . 13 | |
12 | elequ1 1997 | . . . . . . . . . . . . 13 | |
13 | 11, 12 | anbi12d 747 | . . . . . . . . . . . 12 |
14 | 10, 13 | anbi12d 747 | . . . . . . . . . . 11 |
15 | 14 | cbvexv 2275 | . . . . . . . . . 10 |
16 | 15 | bibi1i 328 | . . . . . . . . 9 |
17 | 8, 16 | syl6bb 276 | . . . . . . . 8 |
18 | 17 | albidv 1849 | . . . . . . 7 |
19 | elequ1 1997 | . . . . . . . . . . . 12 | |
20 | 19 | anbi1d 741 | . . . . . . . . . . 11 |
21 | elequ1 1997 | . . . . . . . . . . . 12 | |
22 | 21 | anbi1d 741 | . . . . . . . . . . 11 |
23 | 20, 22 | anbi12d 747 | . . . . . . . . . 10 |
24 | 23 | exbidv 1850 | . . . . . . . . 9 |
25 | equequ1 1952 | . . . . . . . . 9 | |
26 | 24, 25 | bibi12d 335 | . . . . . . . 8 |
27 | 26 | cbvalv 2273 | . . . . . . 7 |
28 | 18, 27 | syl6bb 276 | . . . . . 6 |
29 | 28 | cbvexv 2275 | . . . . 5 |
30 | 29 | imbi2i 326 | . . . 4 |
31 | 30 | 2albii 1748 | . . 3 |
32 | 31 | exbii 1774 | . 2 |
33 | 6, 32 | mpbir 221 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-reg 8497 ax-ac 9281 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 df-fr 5073 |
This theorem is referenced by: (None) |
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