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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 5573 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉) | |
2 | opeq1 4797 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → 〈𝑝, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
3 | 2 | eqeq2d 2832 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐴 = 〈𝑝, 𝑧〉 ↔ 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
4 | 3 | rexbidv 3297 | . . 3 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
5 | 4 | rexxp 5707 | . 2 ⊢ (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
6 | df-ot 4568 | . . . . . . 7 ⊢ 〈𝑥, 𝑦, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉 | |
7 | 6 | eqcomi 2830 | . . . . . 6 ⊢ 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈𝑥, 𝑦, 𝑧〉 |
8 | 7 | eqeq2i 2834 | . . . . 5 ⊢ (𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
9 | 8 | rexbii 3247 | . . . 4 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
10 | 9 | rexbii 3247 | . . 3 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
11 | 10 | rexbii 3247 | . 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 298 | 1 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∃wrex 3139 〈cop 4565 〈cotp 4567 × cxp 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-ot 4568 df-iun 4914 df-opab 5121 df-xp 5555 df-rel 5556 |
This theorem is referenced by: (None) |
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