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Mirrors > Home > MPE Home > Th. List > Mathboxes > elxpxp | Structured version Visualization version GIF version |
Description: Membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.) |
Ref | Expression |
---|---|
elxpxp | ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp2 5552 | . 2 ⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉) | |
2 | opeq1 4764 | . . . . 5 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → 〈𝑝, 𝑧〉 = 〈〈𝑥, 𝑦〉, 𝑧〉) | |
3 | 2 | eqeq2d 2769 | . . . 4 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝐴 = 〈𝑝, 𝑧〉 ↔ 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
4 | 3 | rexbidv 3221 | . . 3 ⊢ (𝑝 = 〈𝑥, 𝑦〉 → (∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉)) |
5 | 4 | rexxp 5688 | . 2 ⊢ (∃𝑝 ∈ (𝐵 × 𝐶)∃𝑧 ∈ 𝐷 𝐴 = 〈𝑝, 𝑧〉 ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈〈𝑥, 𝑦〉, 𝑧〉) |
6 | 1, 5 | bitri 278 | 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 3071 〈cop 4531 × cxp 5526 |
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 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-iun 4888 df-opab 5099 df-xp 5534 df-rel 5535 |
This theorem is referenced by: elxpxpss 33215 ralxp3f 33216 frpoins3xp3g 33345 poxp3 33363 xpord3pred 33365 sexp3 33366 no3indslem 33697 |
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