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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 5543 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉) | |
2 | opeq1 4763 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → 〈𝑝, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
3 | 2 | eqeq2d 2809 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐴 = 〈𝑝, 𝑧〉 ↔ 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
4 | 3 | rexbidv 3256 | . . 3 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
5 | 4 | rexxp 5677 | . 2 ⊢ (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
6 | df-ot 4534 | . . . . . . 7 ⊢ 〈𝑥, 𝑦, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉 | |
7 | 6 | eqcomi 2807 | . . . . . 6 ⊢ 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈𝑥, 𝑦, 𝑧〉 |
8 | 7 | eqeq2i 2811 | . . . . 5 ⊢ (𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
9 | 8 | rexbii 3210 | . . . 4 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
10 | 9 | rexbii 3210 | . . 3 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
11 | 10 | rexbii 3210 | . 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 300 | 1 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 〈cop 4531 〈cotp 4533 × cxp 5517 |
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 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-ot 4534 df-iun 4883 df-opab 5093 df-xp 5525 df-rel 5526 |
This theorem is referenced by: (None) |
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