Theorem eqelssd 3116

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 114	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
4	3	ssrdv 3103	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
5	1, 4	eqssd 3114	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ⊆ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  fiuni  6866  ennnfonelemrn  11937  ennnfonelemdm  11938  unirnblps  12596  unirnbl  12597  dvidlemap  12834  dviaddf  12843  dvimulf  12844  dvcj  12847  dvrecap  12851