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Theorem sspw12 4336
 Description: A set is a subset of cardinal one iff it is the unit power class of some other set. (Contributed by SF, 17-Mar-2015.)
Hypothesis
Ref Expression
sspw12.1 A V
Assertion
Ref Expression
sspw12 (A 1cx A = 1x)
Distinct variable group:   x,A

Proof of Theorem sspw12
StepHypRef Expression
1 eqpw1uni 4330 . . 3 (A 1cA = 1A)
2 sspw12.1 . . . . 5 A V
32uniex 4317 . . . 4 A V
4 pw1eq 4143 . . . . 5 (x = A1x = 1A)
54eqeq2d 2364 . . . 4 (x = A → (A = 1xA = 1A))
63, 5spcev 2946 . . 3 (A = 1Ax A = 1x)
71, 6syl 15 . 2 (A 1cx A = 1x)
8 pw1ss1c 4158 . . . 4 1x 1c
9 sseq1 3292 . . . 4 (A = 1x → (A 1c1x 1c))
108, 9mpbiri 224 . . 3 (A = 1xA 1c)
1110exlimiv 1634 . 2 (x A = 1xA 1c)
127, 11impbii 180 1 (A 1cx A = 1x)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃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:  ce0lenc1  6239
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