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Theorem pwsnOLD 4817
Description: Obsolete version of pwsn 4816 as of 14-Apr-2024. Note that the proof is essentially the same once one inlines sssn 4743 in the proof of pwsn 4816. (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 3939 . . . . . . . . 9 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}))
2 velsn 4566 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
32imbi2i 339 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {𝐴}) ↔ (𝑦𝑥𝑦 = 𝐴))
43albii 1821 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}) ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
51, 4bitri 278 . . . . . . . 8 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
6 neq0 4292 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1894 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
86, 7syl5bi 245 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
9 dfclel 2897 . . . . . . . . . . 11 (𝐴𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦𝑥))
10 exancom 1862 . . . . . . . . . . 11 (∃𝑦(𝑦 = 𝐴𝑦𝑥) ↔ ∃𝑦(𝑦𝑥𝑦 = 𝐴))
119, 10bitr2i 279 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = 𝐴) ↔ 𝐴𝑥)
12 snssi 4725 . . . . . . . . . 10 (𝐴𝑥 → {𝐴} ⊆ 𝑥)
1311, 12sylbi 220 . . . . . . . . 9 (∃𝑦(𝑦𝑥𝑦 = 𝐴) → {𝐴} ⊆ 𝑥)
148, 13syl6 35 . . . . . . . 8 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
155, 14sylbi 220 . . . . . . 7 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
1615anc2li 559 . . . . . 6 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥)))
17 eqss 3968 . . . . . 6 (𝑥 = {𝐴} ↔ (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥))
1816, 17syl6ibr 255 . . . . 5 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → 𝑥 = {𝐴}))
1918orrd 860 . . . 4 (𝑥 ⊆ {𝐴} → (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
20 0ss 4333 . . . . . 6 ∅ ⊆ {𝐴}
21 sseq1 3978 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {𝐴} ↔ ∅ ⊆ {𝐴}))
2220, 21mpbiri 261 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {𝐴})
23 eqimss 4009 . . . . 5 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
2422, 23jaoi 854 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
2519, 24impbii 212 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2625abbii 2889 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
27 df-pw 4524 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
28 dfpr2 4569 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
2926, 27, 283eqtr4i 2857 1 𝒫 {𝐴} = {∅, {𝐴}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  wal 1536   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550  {cpr 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553
This theorem is referenced by: (None)
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