Theorem eqelssd 3189

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 3176	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
5	1, 4	eqssd 3187	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160   ⊆ wss 3144

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-in 3150  df-ss 3157

This theorem is referenced by:  fiuni  7006  ennnfonelemrn  12469  ennnfonelemdm  12470  unirnblps  14374  unirnbl  14375  dvidlemap  14612  dviaddf  14621  dvimulf  14622  dvcj  14625  dvrecap  14629