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Mirrors > Home > MPE Home > Th. List > Mathboxes > sprsymrelfolem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for sprsymrelfo 42271. (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 6342 | . . 3 ⊢ (Pairs‘𝑉) ∈ V | |
3 | ssrab2 3836 | . . 3 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⊆ (Pairs‘𝑉) | |
4 | 2, 3 | elpwi2 4960 | . 2 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ∈ 𝒫 (Pairs‘𝑉) |
5 | 1, 4 | eqeltri 2846 | 1 ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ∀wral 3061 {crab 3065 Vcvv 3351 𝒫 cpw 4297 {cpr 4318 class class class wbr 4786 ‘cfv 6031 Pairscspr 42251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-pw 4299 df-sn 4317 df-pr 4319 df-uni 4575 df-iota 5994 df-fv 6039 |
This theorem is referenced by: sprsymrelfo 42271 |
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