Theorem sprssspr 48085

Description: The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)

Assertion

Ref	Expression
sprssspr	⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Distinct variable group:   𝑉,𝑎,𝑏,𝑝

Proof of Theorem sprssspr

Step	Hyp	Ref	Expression
1		sprval 48083	. . 3 ⊢ (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}})
2		r2ex 3202	. . . . . . 7 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 = {𝑎, 𝑏}))
3		simpr 489	. . . . . . . 8 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏})
4	3	2eximi 1859	. . . . . . 7 ⊢ (∃𝑎∃𝑏((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 = {𝑎, 𝑏}) → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
5	2, 4	sylbi 220	. . . . . 6 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
6	5	ax-gen 1818	. . . . 5 ⊢ ∀𝑝(∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏})
7	6	a1i 11	. . . 4 ⊢ (𝑉 ∈ V → ∀𝑝(∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
8		ss2ab 4017	. . . 4 ⊢ ({𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∀𝑝(∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}))
9	7, 8	sylibr 237	. . 3 ⊢ (𝑉 ∈ V → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}} ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
10	1, 9	eqsstrd 3973	. 2 ⊢ (𝑉 ∈ V → (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
11		fvprc 6863	. . 3 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
12		0ss 4357	. . . 4 ⊢ ∅ ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}
13	12	a1i 11	. . 3 ⊢ (¬ 𝑉 ∈ V → ∅ ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
14	11, 13	eqsstrd 3973	. 2 ⊢ (¬ 𝑉 ∈ V → (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}})
15	10, 14	pm2.61i 184	1 ⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  {cpr 4587  ‘cfv 6525  Pairscspr 48081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-spr 48082

This theorem is referenced by:  spr0el  48086