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Mirrors > Home > NFE Home > Th. List > elpw1 | GIF version |
Description: Membership in a unit power class. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
elpw1 | ⊢ (A ∈ ℘1B ↔ ∃x ∈ B A = {x}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pw1 4137 | . . . 4 ⊢ ℘1B = (℘B ∩ 1c) | |
2 | 1 | eleq2i 2417 | . . 3 ⊢ (A ∈ ℘1B ↔ A ∈ (℘B ∩ 1c)) |
3 | elin 3219 | . . 3 ⊢ (A ∈ (℘B ∩ 1c) ↔ (A ∈ ℘B ∧ A ∈ 1c)) | |
4 | 2, 3 | bitri 240 | . 2 ⊢ (A ∈ ℘1B ↔ (A ∈ ℘B ∧ A ∈ 1c)) |
5 | el1c 4139 | . . . . 5 ⊢ (A ∈ 1c ↔ ∃x A = {x}) | |
6 | 5 | anbi2i 675 | . . . 4 ⊢ ((A ∈ ℘B ∧ A ∈ 1c) ↔ (A ∈ ℘B ∧ ∃x A = {x})) |
7 | 19.42v 1905 | . . . 4 ⊢ (∃x(A ∈ ℘B ∧ A = {x}) ↔ (A ∈ ℘B ∧ ∃x A = {x})) | |
8 | 6, 7 | bitr4i 243 | . . 3 ⊢ ((A ∈ ℘B ∧ A ∈ 1c) ↔ ∃x(A ∈ ℘B ∧ A = {x})) |
9 | eleq1 2413 | . . . . . . 7 ⊢ (A = {x} → (A ∈ ℘B ↔ {x} ∈ ℘B)) | |
10 | snex 4111 | . . . . . . . . 9 ⊢ {x} ∈ V | |
11 | 10 | elpw 3728 | . . . . . . . 8 ⊢ ({x} ∈ ℘B ↔ {x} ⊆ B) |
12 | vex 2862 | . . . . . . . . 9 ⊢ x ∈ V | |
13 | 12 | snss 3838 | . . . . . . . 8 ⊢ (x ∈ B ↔ {x} ⊆ B) |
14 | 11, 13 | bitr4i 243 | . . . . . . 7 ⊢ ({x} ∈ ℘B ↔ x ∈ B) |
15 | 9, 14 | syl6bb 252 | . . . . . 6 ⊢ (A = {x} → (A ∈ ℘B ↔ x ∈ B)) |
16 | 15 | pm5.32ri 619 | . . . . 5 ⊢ ((A ∈ ℘B ∧ A = {x}) ↔ (x ∈ B ∧ A = {x})) |
17 | 16 | exbii 1582 | . . . 4 ⊢ (∃x(A ∈ ℘B ∧ A = {x}) ↔ ∃x(x ∈ B ∧ A = {x})) |
18 | df-rex 2620 | . . . 4 ⊢ (∃x ∈ B A = {x} ↔ ∃x(x ∈ B ∧ A = {x})) | |
19 | 17, 18 | bitr4i 243 | . . 3 ⊢ (∃x(A ∈ ℘B ∧ A = {x}) ↔ ∃x ∈ B A = {x}) |
20 | 8, 19 | bitri 240 | . 2 ⊢ ((A ∈ ℘B ∧ A ∈ 1c) ↔ ∃x ∈ B A = {x}) |
21 | 4, 20 | bitri 240 | 1 ⊢ (A ∈ ℘1B ↔ ∃x ∈ B A = {x}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∩ cin 3208 ⊆ wss 3257 ℘cpw 3722 {csn 3737 1cc1c 4134 ℘1cpw1 4135 |
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 ax-nin 4078 ax-sn 4087 |
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-ne 2518 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-1c 4136 df-pw1 4137 |
This theorem is referenced by: elpw12 4145 snelpw1 4146 elpw11c 4147 elpw121c 4148 elpw131c 4149 elpw141c 4150 elpw151c 4151 elpw161c 4152 elpw171c 4153 elpw181c 4154 elpw191c 4155 elpw1101c 4156 elpw1111c 4157 pw1un 4163 pw1in 4164 pw1sn 4165 pw1disj 4167 df1c2 4168 dfpw12 4301 unipw1 4325 vfinspss 4551 vfinncsp 4554 elimapw1 4944 elimapw12 4945 elimapw13 4946 dmsi 5519 pw1fnex 5852 enpw1 6062 enpw1pw 6075 nenpw1pwlem2 6085 scancan 6331 |
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