Theorem snsssn 3787

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 3168	. . 3 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}))
2		velsn 3635	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		velsn 3635	. . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	2, 3	imbi12i 239	. . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 → 𝑥 = 𝐵))
5	4	albii 1481	. . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵))
6	1, 5	bitri 184	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵))
7		sneqr.1	. . 3 ⊢ 𝐴 ∈ V
8		sbceqal 3041	. . 3 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
9	7, 8	ax-mp 5	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)
10	6, 9	sylbi 121	1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3153  {csn 3618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-in 3159  df-ss 3166  df-sn 3624

This theorem is referenced by: (None)