Theorem difsnpssim 3507

Description: (𝐵 ∖ {𝐴}) is a proper subclass of 𝐵 if 𝐴 is a member of 𝐵. In classical logic, the converse holds as well. (Contributed by Jim Kingdon, 9-Aug-2018.)

Assertion

Ref	Expression
difsnpssim	⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Proof of Theorem difsnpssim

Step	Hyp	Ref	Expression
1		notnot 559	. 2 ⊢ (𝐴 ∈ 𝐵 → ¬ ¬ 𝐴 ∈ 𝐵)
2		difss 3070	. . . 4 ⊢ (𝐵 ∖ {𝐴}) ⊆ 𝐵
3	2	biantrur 287	. . 3 ⊢ ((𝐵 ∖ {𝐴}) ≠ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
4		difsnb 3506	. . . 4 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) = 𝐵)
5	4	necon3bbii 2242	. . 3 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ≠ 𝐵)
6		df-pss 2933	. . 3 ⊢ ((𝐵 ∖ {𝐴}) ⊊ 𝐵 ↔ ((𝐵 ∖ {𝐴}) ⊆ 𝐵 ∧ (𝐵 ∖ {𝐴}) ≠ 𝐵))
7	3, 5, 6	3bitr4i 201	. 2 ⊢ (¬ ¬ 𝐴 ∈ 𝐵 ↔ (𝐵 ∖ {𝐴}) ⊊ 𝐵)
8	1, 7	sylib 127	1 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∖ {𝐴}) ⊊ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1393   ≠ wne 2204   ∖ cdif 2914   ⊆ wss 2917   ⊊ wpss 2918  {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-pss 2933  df-sn 3381

This theorem is referenced by: (None)