| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > elxpi | Structured version Visualization version GIF version | ||
| Description: Membership in a Cartesian product. Uses fewer axioms than elxp 5708. (Contributed by NM, 4-Jul-1994.) |
| Ref | Expression |
|---|---|
| elxpi | ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq1 2741 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = 〈𝑥, 𝑦〉 ↔ 𝐴 = 〈𝑥, 𝑦〉)) | |
| 2 | 1 | anbi1d 631 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))) |
| 3 | 2 | 2exbidv 1924 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))) |
| 4 | df-xp 5691 | . . . 4 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
| 5 | df-opab 5206 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} | |
| 6 | 4, 5 | eqtri 2765 | . . 3 ⊢ (𝐵 × 𝐶) = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))} |
| 7 | 3, 6 | elab2g 3680 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))) |
| 8 | 7 | ibi 267 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 〈cop 4632 {copab 5205 × cxp 5683 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: xpdifid 6188 opreuopreu 8059 djuunxp 9961 rngqiprngimfo 21311 fsumdvdsmul 27238 fmla0xp 35388 mnringmulrcld 44247 rrx2plordisom 48644 |
| Copyright terms: Public domain | W3C validator |