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| Mirrors > Home > MPE Home > Th. List > el2xpss | Structured version Visualization version GIF version | ||
| Description: Version of elrel 5808 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 3978 | . . 3 ⊢ ((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷)) | |
| 2 | 1 | ancoms 458 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → 𝐴 ∈ ((𝐵 × 𝐶) × 𝐷)) | 
| 3 | el2xptp 8060 | . . 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 217 | . 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 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 〈cotp 4634 × 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-ot 4635 df-iun 4993 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: frxp3 8176 | 
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