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Theorem sspw1 4335
 Description: A condition for being a subclass of a unit power class. Corollary 2 of theorem IX.6.14 of [Rosser] p. 255. (Contributed by SF, 3-Feb-2015.)
Hypothesis
Ref Expression
sspw1.1 A V
Assertion
Ref Expression
sspw1 (A 1Bx(x B A = 1x))
Distinct variable groups:   x,A   x,B

Proof of Theorem sspw1
StepHypRef Expression
1 uniss 3912 . . . 4 (A 1BA 1B)
2 unipw1 4325 . . . 4 1B = B
31, 2syl6sseq 3317 . . 3 (A 1BA B)
4 pw1ss1c 4158 . . . . 5 1B 1c
5 sstr 3280 . . . . 5 ((A 1B 1B 1c) → A 1c)
64, 5mpan2 652 . . . 4 (A 1BA 1c)
7 eqpw1uni 4330 . . . 4 (A 1cA = 1A)
86, 7syl 15 . . 3 (A 1BA = 1A)
9 sspw1.1 . . . . 5 A V
109uniex 4317 . . . 4 A V
11 sseq1 3292 . . . . 5 (x = A → (x BA B))
12 pw1eq 4143 . . . . . 6 (x = A1x = 1A)
1312eqeq2d 2364 . . . . 5 (x = A → (A = 1xA = 1A))
1411, 13anbi12d 691 . . . 4 (x = A → ((x B A = 1x) ↔ (A B A = 1A)))
1510, 14spcev 2946 . . 3 ((A B A = 1A) → x(x B A = 1x))
163, 8, 15syl2anc 642 . 2 (A 1Bx(x B A = 1x))
17 pw1ss 4169 . . . . 5 (x B1x 1B)
18 sseq1 3292 . . . . 5 (A = 1x → (A 1B1x 1B))
1917, 18syl5ibr 212 . . . 4 (A = 1x → (x BA 1B))
2019impcom 419 . . 3 ((x B A = 1x) → A 1B)
2120exlimiv 1634 . 2 (x(x B A = 1x) → A 1B)
2216, 21impbii 180 1 (A 1Bx(x B A = 1x))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  ∪cuni 3891  1cc1c 4134  ℘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-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  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-ral 2619  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-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193 This theorem is referenced by:  vfinspsslem1  4550  pw1fnf1o  5855  enpw1pw  6075  ce0addcnnul  6179
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