Theorem ssextss 1864

Description: An extensionality-like principle defining subclass in terms of subsets.

Assertion

Ref	Expression
ssextss	⊢ (A ⊆ B ↔ ∀x(x ⊆ A → x ⊆ B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem ssextss

Step	Hyp	Ref	Expression
1		sspwb 1863	. 2 ⊢ (A ⊆ B ↔ ℘A ⊆ ℘B)
2		dfss2 1497	. 2 ⊢ (℘A ⊆ ℘B ↔ ∀x(x ∈ ℘A → x ∈ ℘B))
3		visset 1350	. . . . 5 ⊢ x ∈ V
4	3	elpw 1801	. . . 4 ⊢ (x ∈ ℘A ↔ x ⊆ A)
5	3	elpw 1801	. . . 4 ⊢ (x ∈ ℘B ↔ x ⊆ B)
6	4, 5	imbi12i 163	. . 3 ⊢ ((x ∈ ℘A → x ∈ ℘B) ↔ (x ⊆ A → x ⊆ B))
7	6	bial 695	. 2 ⊢ (∀x(x ∈ ℘A → x ∈ ℘B) ↔ ∀x(x ⊆ A → x ⊆ B))
8	1, 2, 7	3bitr 155	1 ⊢ (A ⊆ B ↔ ∀x(x ⊆ A → x ⊆ B))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   ∈ wcel 1092   ⊆ wss 1487  ℘cpw 1798

This theorem is referenced by:  ssext 1865  nssss 1866

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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