Theorem sspw1or2 7402

Description: The set of subsets of a given set with one or two elements can be expressed as elements of the power set or as inhabited elements of the power set. (Contributed by Jim Kingdon, 31-Mar-2026.)

Assertion

Ref	Expression
sspw1or2	⊢ {𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} = {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}

Distinct variable groups:   𝑉,𝑠   𝑗,𝑠,𝑥

Allowed substitution hints:   𝑉(𝑥,𝑗)

Proof of Theorem sspw1or2

Step	Hyp	Ref	Expression
1		elequ2 2207	. . . . . 6 ⊢ (𝑠 = 𝑥 → (𝑗 ∈ 𝑠 ↔ 𝑗 ∈ 𝑥))
2	1	exbidv 1873	. . . . 5 ⊢ (𝑠 = 𝑥 → (∃𝑗 𝑗 ∈ 𝑠 ↔ ∃𝑗 𝑗 ∈ 𝑥))
3	2	elrab 2962	. . . 4 ⊢ (𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ↔ (𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥))
4	3	anbi1i 458	. . 3 ⊢ ((𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)) ↔ ((𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥) ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)))
5		en1m 6978	. . . . . 6 ⊢ (𝑥 ≈ 1o → ∃𝑗 𝑗 ∈ 𝑥)
6		en2m 6998	. . . . . 6 ⊢ (𝑥 ≈ 2o → ∃𝑗 𝑗 ∈ 𝑥)
7	5, 6	jaoi 723	. . . . 5 ⊢ ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) → ∃𝑗 𝑗 ∈ 𝑥)
8	7	biantrud 304	. . . 4 ⊢ ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) → (𝑥 ∈ 𝒫 𝑉 ↔ (𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥)))
9	8	pm5.32ri 455	. . 3 ⊢ ((𝑥 ∈ 𝒫 𝑉 ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)) ↔ ((𝑥 ∈ 𝒫 𝑉 ∧ ∃𝑗 𝑗 ∈ 𝑥) ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)))
10	4, 9	bitr4i 187	. 2 ⊢ ((𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)) ↔ (𝑥 ∈ 𝒫 𝑉 ∧ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)))
11	10	rabbia2 2787	1 ⊢ {𝑥 ∈ {𝑠 ∈ 𝒫 𝑉 ∣ ∃𝑗 𝑗 ∈ 𝑠} ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} = {𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}

Syntax hints:   ∧ wa 104   ∨ wo 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {crab 2514  𝒫 cpw 3652   class class class wbr 4088  1_oc1o 6574  2_oc2o 6575   ≈ cen 6906

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-en 6909

This theorem is referenced by:  subupgr  16123