Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-opabssvv | Structured version Visualization version GIF version |
Description: A variant of relopabiv 5686 (which could be proved from it, similarly to relxp 5566 from xpss 5564). (Contributed by BJ, 28-Dec-2023.) |
Ref | Expression |
---|---|
bj-opabssvv | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3494 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | vex 3494 | . . . . 5 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | pm3.2i 473 | . . . 4 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V)) |
5 | 4 | ssopab2i 5430 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} |
6 | df-xp 5554 | . 2 ⊢ (V × V) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} | |
7 | 5, 6 | sseqtrri 3997 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} ⊆ (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2113 Vcvv 3491 ⊆ wss 3929 {copab 5121 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-in 3936 df-ss 3945 df-opab 5122 df-xp 5554 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |