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| Mirrors > Home > MPE Home > Th. List > el2xptp | Structured version Visualization version GIF version | ||
| Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.) | 
| Ref | Expression | 
|---|---|
| el2xptp | ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elxp2 5709 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉) | |
| 2 | opeq1 4873 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → 〈𝑝, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
| 3 | 2 | eqeq2d 2748 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐴 = 〈𝑝, 𝑧〉 ↔ 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) | 
| 4 | 3 | rexbidv 3179 | . . 3 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) | 
| 5 | 4 | rexxp 5853 | . 2 ⊢ (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) | 
| 6 | df-ot 4635 | . . . . . . 7 ⊢ 〈𝑥, 𝑦, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉 | |
| 7 | 6 | eqcomi 2746 | . . . . . 6 ⊢ 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈𝑥, 𝑦, 𝑧〉 | 
| 8 | 7 | eqeq2i 2750 | . . . . 5 ⊢ (𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | 
| 9 | 8 | rexbii 3094 | . . . 4 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | 
| 10 | 9 | rexbii 3094 | . . 3 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | 
| 11 | 10 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | 
| 12 | 1, 5, 11 | 3bitri 297 | 1 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 〈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: el2xpss 8062 ralxp3f 8162 frpoins3xp3g 8166 poxp3 8175 xpord3pred 8177 sexp3 8178 | 
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