Theorem eqelssd 3247

Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
eqelssd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
eqelssd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
eqelssd	⊢ (𝜑 → 𝐴 = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem eqelssd

Step	Hyp	Ref	Expression
1		eqelssd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		eqelssd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
3	2	ex 115	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
4	3	ssrdv 3234	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
5	1, 4	eqssd 3245	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202   ⊆ wss 3201

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  fiuni  7237  ennnfonelemrn  13120  ennnfonelemdm  13121  unirnblps  15233  unirnbl  15234  dvidlemap  15502  dvidrelem  15503  dvidsslem  15504  dviaddf  15516  dvimulf  15517  dvcj  15520  dvrecap  15524