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Mirrors > Home > MPE Home > Th. List > el2xpss | Structured version Visualization version GIF version |
Description: Version of elrel 5796 for triple Cartesian products. (Contributed by Scott Fenton, 1-Feb-2025.) |
Ref | Expression |
---|---|
el2xpss | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3976 | . . 3 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷)) | |
2 | 1 | ancoms 459 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷)) |
3 | el2xptp 8017 | . . 3 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) | |
4 | rexex 3076 | . . . . . . 7 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) | |
5 | 4 | reximi 3084 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) |
6 | rexex 3076 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) |
8 | 7 | reximi 3084 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) |
9 | rexex 3076 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) |
11 | 3, 10 | sylbi 216 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) |
12 | 2, 11 | syl 17 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃wrex 3070 ⊆ wss 3947 ⟨cotp 4635 × cxp 5673 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-ot 4636 df-iun 4998 df-opab 5210 df-xp 5681 df-rel 5682 |
This theorem is referenced by: frxp3 8133 |
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