Theorem prssd 3751

Description: Deduction version of prssi 3750: A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
prssd.1	⊢ (𝜑 → 𝐴 ∈ 𝐶)
prssd.2	⊢ (𝜑 → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
prssd	⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)

Proof of Theorem prssd

Step	Hyp	Ref	Expression
1		prssd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
2		prssd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
3		prssi 3750	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ⊆ 𝐶)
4	1, 2, 3	syl2anc 411	1 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2148   ⊆ wss 3129  {cpr 3593

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599

This theorem is referenced by:  0idnsgd  13007