Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5581 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉) | |
2 | opeq1 4805 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → 〈𝑝, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
3 | 2 | eqeq2d 2834 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐴 = 〈𝑝, 𝑧〉 ↔ 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
4 | 3 | rexbidv 3299 | . . 3 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
5 | 4 | rexxp 5715 | . 2 ⊢ (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
6 | df-ot 4578 | . . . . . . 7 ⊢ 〈𝑥, 𝑦, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉 | |
7 | 6 | eqcomi 2832 | . . . . . 6 ⊢ 〈〈𝑥, 𝑦〉, 𝑧〉 = 〈𝑥, 𝑦, 𝑧〉 |
8 | 7 | eqeq2i 2836 | . . . . 5 ⊢ (𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
9 | 8 | rexbii 3249 | . . . 4 ⊢ (∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
10 | 9 | rexbii 3249 | . . 3 ⊢ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
11 | 10 | rexbii 3249 | . 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 299 | 1 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 〈cop 4575 〈cotp 4577 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-ot 4578 df-iun 4923 df-opab 5131 df-xp 5563 df-rel 5564 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |