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Theorem rabsspr 32520
Description: Conditions for a restricted class abstraction to be a subset of an unordered pair. (Contributed by Thierry Arnoux, 6-Jul-2025.)
Assertion
Ref Expression
rabsspr ({𝑥𝑉𝜑} ⊆ {𝑋, 𝑌} ↔ ∀𝑥𝑉 (𝜑 → (𝑥 = 𝑋𝑥 = 𝑌)))
Distinct variable groups:   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rabsspr
StepHypRef Expression
1 df-rab 3437 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 dfpr2 4646 . . 3 {𝑋, 𝑌} = {𝑥 ∣ (𝑥 = 𝑋𝑥 = 𝑌)}
31, 2sseq12i 4014 . 2 ({𝑥𝑉𝜑} ⊆ {𝑋, 𝑌} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥 ∣ (𝑥 = 𝑋𝑥 = 𝑌)})
4 ss2ab 4062 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥 ∣ (𝑥 = 𝑋𝑥 = 𝑌)} ↔ ∀𝑥((𝑥𝑉𝜑) → (𝑥 = 𝑋𝑥 = 𝑌)))
5 impexp 450 . . . 4 (((𝑥𝑉𝜑) → (𝑥 = 𝑋𝑥 = 𝑌)) ↔ (𝑥𝑉 → (𝜑 → (𝑥 = 𝑋𝑥 = 𝑌))))
65albii 1819 . . 3 (∀𝑥((𝑥𝑉𝜑) → (𝑥 = 𝑋𝑥 = 𝑌)) ↔ ∀𝑥(𝑥𝑉 → (𝜑 → (𝑥 = 𝑋𝑥 = 𝑌))))
7 df-ral 3062 . . 3 (∀𝑥𝑉 (𝜑 → (𝑥 = 𝑋𝑥 = 𝑌)) ↔ ∀𝑥(𝑥𝑉 → (𝜑 → (𝑥 = 𝑋𝑥 = 𝑌))))
86, 7bitr4i 278 . 2 (∀𝑥((𝑥𝑉𝜑) → (𝑥 = 𝑋𝑥 = 𝑌)) ↔ ∀𝑥𝑉 (𝜑 → (𝑥 = 𝑋𝑥 = 𝑌)))
93, 4, 83bitri 297 1 ({𝑥𝑉𝜑} ⊆ {𝑋, 𝑌} ↔ ∀𝑥𝑉 (𝜑 → (𝑥 = 𝑋𝑥 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  wal 1538   = wceq 1540  wcel 2108  {cab 2714  wral 3061  {crab 3436  wss 3951  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629
This theorem is referenced by:  constrfin  33787
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