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Mirrors > Home > NFE Home > Th. List > dmsnn0 | GIF version |
Description: The domain of a singleton is nonzero iff the singleton argument is a set. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Scott Fenton, 19-Apr-2021.) |
Ref | Expression |
---|---|
dmsnn0 | ⊢ (A ∈ V ↔ dom {A} ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldm2 4899 | . . . 4 ⊢ (x ∈ dom {A} ↔ ∃y〈x, y〉 ∈ {A}) | |
2 | vex 2862 | . . . . . . . 8 ⊢ x ∈ V | |
3 | vex 2862 | . . . . . . . 8 ⊢ y ∈ V | |
4 | 2, 3 | opex 4588 | . . . . . . 7 ⊢ 〈x, y〉 ∈ V |
5 | 4 | elsnc 3756 | . . . . . 6 ⊢ (〈x, y〉 ∈ {A} ↔ 〈x, y〉 = A) |
6 | eqcom 2355 | . . . . . 6 ⊢ (〈x, y〉 = A ↔ A = 〈x, y〉) | |
7 | 5, 6 | bitri 240 | . . . . 5 ⊢ (〈x, y〉 ∈ {A} ↔ A = 〈x, y〉) |
8 | 7 | exbii 1582 | . . . 4 ⊢ (∃y〈x, y〉 ∈ {A} ↔ ∃y A = 〈x, y〉) |
9 | 1, 8 | bitri 240 | . . 3 ⊢ (x ∈ dom {A} ↔ ∃y A = 〈x, y〉) |
10 | 9 | exbii 1582 | . 2 ⊢ (∃x x ∈ dom {A} ↔ ∃x∃y A = 〈x, y〉) |
11 | n0 3559 | . 2 ⊢ (dom {A} ≠ ∅ ↔ ∃x x ∈ dom {A}) | |
12 | opeqexb 4620 | . 2 ⊢ (A ∈ V ↔ ∃x∃y A = 〈x, y〉) | |
13 | 10, 11, 12 | 3bitr4ri 269 | 1 ⊢ (A ∈ V ↔ dom {A} ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 Vcvv 2859 ∅c0 3550 {csn 3737 〈cop 4561 dom cdm 4772 |
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-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: rnsnn0 5065 |
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