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Mirrors > Home > MPE Home > Th. List > Mathboxes > sprsymrelfolem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for sprsymrelfo 43536. (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 6676 | . . 3 ⊢ (Pairs‘𝑉) ∈ V | |
3 | ssrab2 4053 | . . 3 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉) | |
4 | 2, 3 | elpwi2 5240 | . 2 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉) |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 Vcvv 3492 𝒫 cpw 4535 {cpr 4559 class class class wbr 5057 ‘cfv 6348 Pairscspr 43516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-pw 4537 df-sn 4558 df-pr 4560 df-uni 4831 df-iota 6307 df-fv 6356 |
This theorem is referenced by: sprsymrelfo 43536 |
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