Theorem snsssn 3532

Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Hypothesis

Ref	Expression
sneqr.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snsssn	⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Proof of Theorem snsssn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 2934	. . 3 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}))
2		velsn 3392	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velsn 3392	. . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	2, 3	imbi12i 228	. . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 → 𝑥 = 𝐵))
5	4	albii 1359	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵))
6	1, 5	bitri 173	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵))
7		sneqr.1	. . 3 ⊢ 𝐴 ∈ V
8		sbceqal 2814	. . 3 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
9	7, 8	ax-mp 7	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)
10	6, 9	sylbi 114	1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  {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-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-v 2559  df-sbc 2765  df-in 2924  df-ss 2931  df-sn 3381

This theorem is referenced by: (None)