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Mirrors > Home > MPE Home > Th. List > Mathboxes > elxpxpss | Structured version Visualization version GIF version |
Description: Version of elrel 5645 for triple cross products. (Contributed by Scott Fenton, 21-Aug-2024.) |
Ref | Expression |
---|---|
elxpxpss | ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3889 | . 2 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷)) | |
2 | elxpxp 33215 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
3 | rexex 3167 | . . . . . . 7 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 → ∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
4 | 3 | reximi 3171 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 → ∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
5 | rexex 3167 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 → ∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 → ∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
7 | 6 | reximi 3171 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 → ∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
8 | rexex 3167 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
10 | 2, 9 | sylbi 220 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
11 | 1, 10 | syl 17 | 1 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃wrex 3071 ⊆ wss 3860 〈cop 4531 × cxp 5526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-iun 4888 df-opab 5099 df-xp 5534 df-rel 5535 |
This theorem is referenced by: frxp3 33365 |
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