Theorem difsnpss 3851

Description: (B ∖ {A}) is a proper subclass of B if and only if A is a member of B. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
difsnpss	⊢ (A ∈ B ↔ (B ∖ {A}) ⊊ B)

Proof of Theorem difsnpss

Step	Hyp	Ref	Expression
1		notnot 282	. 2 ⊢ (A ∈ B ↔ ¬ ¬ A ∈ B)
2		difss 3393	. . . 4 ⊢ (B ∖ {A}) ⊆ B
3	2	biantrur 492	. . 3 ⊢ ((B ∖ {A}) ≠ B ↔ ((B ∖ {A}) ⊆ B ∧ (B ∖ {A}) ≠ B))
4		difsnb 3850	. . . 4 ⊢ (¬ A ∈ B ↔ (B ∖ {A}) = B)
5	4	necon3bbii 2547	. . 3 ⊢ (¬ ¬ A ∈ B ↔ (B ∖ {A}) ≠ B)
6		df-pss 3261	. . 3 ⊢ ((B ∖ {A}) ⊊ B ↔ ((B ∖ {A}) ⊆ B ∧ (B ∖ {A}) ≠ B))
7	3, 5, 6	3bitr4i 268	. 2 ⊢ (¬ ¬ A ∈ B ↔ (B ∖ {A}) ⊊ B)
8	1, 7	bitri 240	1 ⊢ (A ∈ B ↔ (B ∖ {A}) ⊊ B)

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ≠ wne 2516   ∖ cdif 3206   ⊆ wss 3257   ⊊ wpss 3258  {csn 3737

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-pss 3261  df-sn 3741

This theorem is referenced by: (None)