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Theorem pwsnOLD 4902
Description: Obsolete version of pwsn 4901 as of 14-Apr-2024. Note that the proof is essentially the same once one inlines sssn 4830 in the proof of pwsn 4901. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwsnOLD 𝒫 {𝐴} = {∅, {𝐴}}

Proof of Theorem pwsnOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3969 . . . . . . . . 9 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}))
2 velsn 4645 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
32imbi2i 336 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {𝐴}) ↔ (𝑦𝑥𝑦 = 𝐴))
43albii 1822 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}) ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
51, 4bitri 275 . . . . . . . 8 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
6 neq0 4346 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1896 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
86, 7biimtrid 241 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
9 dfclel 2812 . . . . . . . . . . 11 (𝐴𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦𝑥))
10 exancom 1865 . . . . . . . . . . 11 (∃𝑦(𝑦 = 𝐴𝑦𝑥) ↔ ∃𝑦(𝑦𝑥𝑦 = 𝐴))
119, 10bitr2i 276 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = 𝐴) ↔ 𝐴𝑥)
12 snssi 4812 . . . . . . . . . 10 (𝐴𝑥 → {𝐴} ⊆ 𝑥)
1311, 12sylbi 216 . . . . . . . . 9 (∃𝑦(𝑦𝑥𝑦 = 𝐴) → {𝐴} ⊆ 𝑥)
148, 13syl6 35 . . . . . . . 8 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
155, 14sylbi 216 . . . . . . 7 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
1615anc2li 557 . . . . . 6 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥)))
17 eqss 3998 . . . . . 6 (𝑥 = {𝐴} ↔ (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥))
1816, 17imbitrrdi 251 . . . . 5 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → 𝑥 = {𝐴}))
1918orrd 862 . . . 4 (𝑥 ⊆ {𝐴} → (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
20 0ss 4397 . . . . . 6 ∅ ⊆ {𝐴}
21 sseq1 4008 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {𝐴} ↔ ∅ ⊆ {𝐴}))
2220, 21mpbiri 258 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {𝐴})
23 eqimss 4041 . . . . 5 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
2422, 23jaoi 856 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
2519, 24impbii 208 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2625abbii 2803 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
27 df-pw 4605 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
28 dfpr2 4648 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
2926, 27, 283eqtr4i 2771 1 𝒫 {𝐴} = {∅, {𝐴}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629  {cpr 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632
This theorem is referenced by: (None)
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