Theorem ssprss 4296

Description: A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)

Assertion

Ref	Expression
ssprss	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))

Proof of Theorem ssprss

Step	Hyp	Ref	Expression
1		prssg 4290	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) ↔ {𝐴, 𝐵} ⊆ {𝐶, 𝐷}))
2		elprg 4144	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐶, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷)))
3		elprg 4144	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ {𝐶, 𝐷} ↔ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷)))
4	2, 3	bi2anan9 913	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ {𝐶, 𝐷} ∧ 𝐵 ∈ {𝐶, 𝐷}) ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))
5	1, 4	bitr3d 269	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴, 𝐵} ⊆ {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∨ 𝐴 = 𝐷) ∧ (𝐵 = 𝐶 ∨ 𝐵 = 𝐷))))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128

This theorem is referenced by:  ssprsseq  4297