Metamath Proof Explorer


Theorem dfacbasgrp

Description: A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988 . (Contributed by Stefan O'Rear, 9-Jul-2015)

Ref Expression
Assertion dfacbasgrp ( CHOICE ↔ ( Base “ Grp ) = ( V ∖ { ∅ } ) )

Proof

Step Hyp Ref Expression
1 dfac10 ( CHOICE ↔ dom card = V )
2 basfn Base Fn V
3 ssv Grp ⊆ V
4 fvelimab ( ( Base Fn V ∧ Grp ⊆ V ) → ( 𝑥 ∈ ( Base “ Grp ) ↔ ∃ 𝑦 ∈ Grp ( Base ‘ 𝑦 ) = 𝑥 ) )
5 2 3 4 mp2an ( 𝑥 ∈ ( Base “ Grp ) ↔ ∃ 𝑦 ∈ Grp ( Base ‘ 𝑦 ) = 𝑥 )
6 eqid ( Base ‘ 𝑦 ) = ( Base ‘ 𝑦 )
7 6 grpbn0 ( 𝑦 ∈ Grp → ( Base ‘ 𝑦 ) ≠ ∅ )
8 neeq1 ( ( Base ‘ 𝑦 ) = 𝑥 → ( ( Base ‘ 𝑦 ) ≠ ∅ ↔ 𝑥 ≠ ∅ ) )
9 7 8 syl5ibcom ( 𝑦 ∈ Grp → ( ( Base ‘ 𝑦 ) = 𝑥𝑥 ≠ ∅ ) )
10 9 rexlimiv ( ∃ 𝑦 ∈ Grp ( Base ‘ 𝑦 ) = 𝑥𝑥 ≠ ∅ )
11 5 10 sylbi ( 𝑥 ∈ ( Base “ Grp ) → 𝑥 ≠ ∅ )
12 11 adantl ( ( dom card = V ∧ 𝑥 ∈ ( Base “ Grp ) ) → 𝑥 ≠ ∅ )
13 vex 𝑥 ∈ V
14 12 13 jctil ( ( dom card = V ∧ 𝑥 ∈ ( Base “ Grp ) ) → ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) )
15 ablgrp ( 𝑥 ∈ Abel → 𝑥 ∈ Grp )
16 15 ssriv Abel ⊆ Grp
17 imass2 ( Abel ⊆ Grp → ( Base “ Abel ) ⊆ ( Base “ Grp ) )
18 16 17 ax-mp ( Base “ Abel ) ⊆ ( Base “ Grp )
19 simprl ( ( dom card = V ∧ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) ) → 𝑥 ∈ V )
20 simpl ( ( dom card = V ∧ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) ) → dom card = V )
21 19 20 eleqtrrd ( ( dom card = V ∧ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) ) → 𝑥 ∈ dom card )
22 simprr ( ( dom card = V ∧ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) ) → 𝑥 ≠ ∅ )
23 isnumbasgrplem3 ( ( 𝑥 ∈ dom card ∧ 𝑥 ≠ ∅ ) → 𝑥 ∈ ( Base “ Abel ) )
24 21 22 23 syl2anc ( ( dom card = V ∧ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) ) → 𝑥 ∈ ( Base “ Abel ) )
25 18 24 sselid ( ( dom card = V ∧ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) ) → 𝑥 ∈ ( Base “ Grp ) )
26 14 25 impbida ( dom card = V → ( 𝑥 ∈ ( Base “ Grp ) ↔ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) ) )
27 eldifsn ( 𝑥 ∈ ( V ∖ { ∅ } ) ↔ ( 𝑥 ∈ V ∧ 𝑥 ≠ ∅ ) )
28 26 27 bitr4di ( dom card = V → ( 𝑥 ∈ ( Base “ Grp ) ↔ 𝑥 ∈ ( V ∖ { ∅ } ) ) )
29 28 eqrdv ( dom card = V → ( Base “ Grp ) = ( V ∖ { ∅ } ) )
30 fvex ( har ‘ 𝑥 ) ∈ V
31 13 30 unex ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∈ V
32 ssun2 ( har ‘ 𝑥 ) ⊆ ( 𝑥 ∪ ( har ‘ 𝑥 ) )
33 harn0 ( 𝑥 ∈ V → ( har ‘ 𝑥 ) ≠ ∅ )
34 13 33 ax-mp ( har ‘ 𝑥 ) ≠ ∅
35 ssn0 ( ( ( har ‘ 𝑥 ) ⊆ ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∧ ( har ‘ 𝑥 ) ≠ ∅ ) → ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ≠ ∅ )
36 32 34 35 mp2an ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ≠ ∅
37 eldifsn ( ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∈ ( V ∖ { ∅ } ) ↔ ( ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∈ V ∧ ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ≠ ∅ ) )
38 31 36 37 mpbir2an ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∈ ( V ∖ { ∅ } )
39 38 a1i ( ( Base “ Grp ) = ( V ∖ { ∅ } ) → ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∈ ( V ∖ { ∅ } ) )
40 id ( ( Base “ Grp ) = ( V ∖ { ∅ } ) → ( Base “ Grp ) = ( V ∖ { ∅ } ) )
41 39 40 eleqtrrd ( ( Base “ Grp ) = ( V ∖ { ∅ } ) → ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∈ ( Base “ Grp ) )
42 isnumbasgrp ( 𝑥 ∈ dom card ↔ ( 𝑥 ∪ ( har ‘ 𝑥 ) ) ∈ ( Base “ Grp ) )
43 41 42 sylibr ( ( Base “ Grp ) = ( V ∖ { ∅ } ) → 𝑥 ∈ dom card )
44 13 a1i ( ( Base “ Grp ) = ( V ∖ { ∅ } ) → 𝑥 ∈ V )
45 43 44 2thd ( ( Base “ Grp ) = ( V ∖ { ∅ } ) → ( 𝑥 ∈ dom card ↔ 𝑥 ∈ V ) )
46 45 eqrdv ( ( Base “ Grp ) = ( V ∖ { ∅ } ) → dom card = V )
47 29 46 impbii ( dom card = V ↔ ( Base “ Grp ) = ( V ∖ { ∅ } ) )
48 1 47 bitri ( CHOICE ↔ ( Base “ Grp ) = ( V ∖ { ∅ } ) )