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Theorem pwsnALT 4537
Description: Alternate proof of pwsn 4536, more direct. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwsnALT 𝒫 {𝐴} = {∅, {𝐴}}

Proof of Theorem pwsnALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3697 . . . . . . . . 9 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}))
2 velsn 4301 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
32imbi2i 325 . . . . . . . . . 10 ((𝑦𝑥𝑦 ∈ {𝐴}) ↔ (𝑦𝑥𝑦 = 𝐴))
43albii 1860 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 ∈ {𝐴}) ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
51, 4bitri 264 . . . . . . . 8 (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦𝑥𝑦 = 𝐴))
6 neq0 4038 . . . . . . . . . 10 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
7 exintr 1932 . . . . . . . . . 10 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
86, 7syl5bi 232 . . . . . . . . 9 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦𝑥𝑦 = 𝐴)))
9 df-clel 2720 . . . . . . . . . . 11 (𝐴𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦𝑥))
10 exancom 1900 . . . . . . . . . . 11 (∃𝑦(𝑦 = 𝐴𝑦𝑥) ↔ ∃𝑦(𝑦𝑥𝑦 = 𝐴))
119, 10bitr2i 265 . . . . . . . . . 10 (∃𝑦(𝑦𝑥𝑦 = 𝐴) ↔ 𝐴𝑥)
12 snssi 4447 . . . . . . . . . 10 (𝐴𝑥 → {𝐴} ⊆ 𝑥)
1311, 12sylbi 207 . . . . . . . . 9 (∃𝑦(𝑦𝑥𝑦 = 𝐴) → {𝐴} ⊆ 𝑥)
148, 13syl6 35 . . . . . . . 8 (∀𝑦(𝑦𝑥𝑦 = 𝐴) → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
155, 14sylbi 207 . . . . . . 7 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
1615anc2li 581 . . . . . 6 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥)))
17 eqss 3724 . . . . . 6 (𝑥 = {𝐴} ↔ (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥))
1816, 17syl6ibr 242 . . . . 5 (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → 𝑥 = {𝐴}))
1918orrd 392 . . . 4 (𝑥 ⊆ {𝐴} → (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
20 0ss 4080 . . . . . 6 ∅ ⊆ {𝐴}
21 sseq1 3732 . . . . . 6 (𝑥 = ∅ → (𝑥 ⊆ {𝐴} ↔ ∅ ⊆ {𝐴}))
2220, 21mpbiri 248 . . . . 5 (𝑥 = ∅ → 𝑥 ⊆ {𝐴})
23 eqimss 3763 . . . . 5 (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
2422, 23jaoi 393 . . . 4 ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
2519, 24impbii 199 . . 3 (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
2625abbii 2841 . 2 {𝑥𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
27 df-pw 4268 . 2 𝒫 {𝐴} = {𝑥𝑥 ⊆ {𝐴}}
28 dfpr2 4303 . 2 {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
2926, 27, 283eqtr4i 2756 1 𝒫 {𝐴} = {∅, {𝐴}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  wal 1594   = wceq 1596  wex 1817  wcel 2103  {cab 2710  wss 3680  c0 4023  𝒫 cpw 4266  {csn 4285  {cpr 4287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-pw 4268  df-sn 4286  df-pr 4288
This theorem is referenced by: (None)
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