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Mirrors > Home > MPE Home > Th. List > Mathboxes > sprsymrelfolem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for sprsymrelfo 46974. (Contributed by AV, 22-Nov-2021.) |
Ref | Expression |
---|---|
sprsymrelfo.q | ⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} |
Ref | Expression |
---|---|
sprsymrelfolem1 | ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sprsymrelfo.q | . 2 ⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} | |
2 | fvex 6909 | . . 3 ⊢ (Pairs‘𝑉) ∈ V | |
3 | ssrab2 4073 | . . 3 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉) | |
4 | 2, 3 | elpwi2 5349 | . 2 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉) |
5 | 1, 4 | eqeltri 2821 | 1 ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 Vcvv 3461 𝒫 cpw 4604 {cpr 4632 class class class wbr 5149 ‘cfv 6549 Pairscspr 46954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-pw 4606 df-sn 4631 df-pr 4633 df-uni 4910 df-iota 6501 df-fv 6557 |
This theorem is referenced by: sprsymrelfo 46974 |
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