Theorem sspw12 4337

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 ⊆ 1c ↔ ∃x A = ℘1x)

Distinct variable group:   x,A

Proof of Theorem sspw12

Step	Hyp	Ref	Expression
1		eqpw1uni 4331	. . 3 ⊢ (A ⊆ 1c → A = ℘1∪A)
2		sspw12.1	. . . . 5 ⊢ A ∈ V
3	2	uniex 4318	. . . 4 ⊢ ∪A ∈ V
4		pw1eq 4144	. . . . 5 ⊢ (x = ∪A → ℘1x = ℘1∪A)
5	4	eqeq2d 2364	. . . 4 ⊢ (x = ∪A → (A = ℘1x ↔ A = ℘1∪A))
6	3, 5	spcev 2947	. . 3 ⊢ (A = ℘1∪A → ∃x A = ℘1x)
7	1, 6	syl 15	. 2 ⊢ (A ⊆ 1c → ∃x A = ℘1x)
8		pw1ss1c 4159	. . . 4 ⊢ ℘1x ⊆ 1c
9		sseq1 3293	. . . 4 ⊢ (A = ℘1x → (A ⊆ 1c ↔ ℘1x ⊆ 1c))
10	8, 9	mpbiri 224	. . 3 ⊢ (A = ℘1x → A ⊆ 1c)
11	10	exlimiv 1634	. 2 ⊢ (∃x A = ℘1x → A ⊆ 1c)
12	7, 11	impbii 180	1 ⊢ (A ⊆ 1c ↔ ∃x A = ℘1x)

Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  ∪cuni 3892  1_cc1c 4135  ℘₁cpw1 4136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-imak 4190  df-p6 4192  df-sik 4193  df-ssetk 4194

This theorem is referenced by:  ce0lenc1  6240