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Mirrors > Home > MPE Home > Th. List > relopabiv | Structured version Visualization version GIF version |
Description: A class of ordered pairs is a relation. For a version without disjoint variable condition, but a longer proof using ax-13 2390, see relopabi 5694. (Contributed by BJ, 22-Jul-2023.) |
Ref | Expression |
---|---|
relopabiv.1 | ⊢ 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
relopabiv | ⊢ Rel 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | pm3.2i 473 | . . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V)) |
5 | 4 | ssopab2i 5437 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} |
6 | relopabiv.1 | . . 3 ⊢ 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
7 | df-xp 5561 | . . 3 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
8 | 5, 6, 7 | 3sstr4i 4010 | . 2 ⊢ 𝐴 ⊆ (V × V) |
9 | df-rel 5562 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
10 | 8, 9 | mpbir 233 | 1 ⊢ Rel 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {copab 5128 × cxp 5553 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-opab 5129 df-xp 5561 df-rel 5562 |
This theorem is referenced by: relfldext 31036 |
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