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Mirrors > Home > NFE Home > Th. List > pw1sn | GIF version |
Description: Compute the unit power class of a singleton. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
pw1sn.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
pw1sn | ⊢ ℘1{A} = {{A}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw1sn.1 | . . . 4 ⊢ A ∈ V | |
2 | sneq 3744 | . . . . 5 ⊢ (y = A → {y} = {A}) | |
3 | 2 | eqeq2d 2364 | . . . 4 ⊢ (y = A → (x = {y} ↔ x = {A})) |
4 | 1, 3 | rexsn 3768 | . . 3 ⊢ (∃y ∈ {A}x = {y} ↔ x = {A}) |
5 | elpw1 4144 | . . 3 ⊢ (x ∈ ℘1{A} ↔ ∃y ∈ {A}x = {y}) | |
6 | elsn 3748 | . . 3 ⊢ (x ∈ {{A}} ↔ x = {A}) | |
7 | 4, 5, 6 | 3bitr4i 268 | . 2 ⊢ (x ∈ ℘1{A} ↔ x ∈ {{A}}) |
8 | 7 | eqriv 2350 | 1 ⊢ ℘1{A} = {{A}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 ∃wrex 2615 Vcvv 2859 {csn 3737 ℘1cpw1 4135 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-3an 936 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-sbc 3047 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: pw1eqadj 4332 ncfinraise 4481 tfinsuc 4498 sfindbl 4530 tc1c 6165 ce0nn 6180 ce2 6192 |
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