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Mirrors > Home > MPE Home > Th. List > pzriprnglem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for pzriprng 21525: An element of 𝐼 is an ordered pair. (Contributed by AV, 18-Mar-2025.) |
Ref | Expression |
---|---|
pzriprng.r | ⊢ 𝑅 = (ℤring ×s ℤring) |
pzriprng.i | ⊢ 𝐼 = (ℤ × {0}) |
Ref | Expression |
---|---|
pzriprnglem3 | ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = 〈𝑥, 0〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pzriprng.i | . . 3 ⊢ 𝐼 = (ℤ × {0}) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ (ℤ × {0})) |
3 | elxp2 5712 | . 2 ⊢ (𝑋 ∈ (ℤ × {0}) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = 〈𝑥, 𝑦〉) | |
4 | 0z 12621 | . . . 4 ⊢ 0 ∈ ℤ | |
5 | opeq2 4878 | . . . . . 6 ⊢ (𝑦 = 0 → 〈𝑥, 𝑦〉 = 〈𝑥, 0〉) | |
6 | 5 | eqeq2d 2745 | . . . . 5 ⊢ (𝑦 = 0 → (𝑋 = 〈𝑥, 𝑦〉 ↔ 𝑋 = 〈𝑥, 0〉)) |
7 | 6 | rexsng 4680 | . . . 4 ⊢ (0 ∈ ℤ → (∃𝑦 ∈ {0}𝑋 = 〈𝑥, 𝑦〉 ↔ 𝑋 = 〈𝑥, 0〉)) |
8 | 4, 7 | ax-mp 5 | . . 3 ⊢ (∃𝑦 ∈ {0}𝑋 = 〈𝑥, 𝑦〉 ↔ 𝑋 = 〈𝑥, 0〉) |
9 | 8 | rexbii 3091 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ {0}𝑋 = 〈𝑥, 𝑦〉 ↔ ∃𝑥 ∈ ℤ 𝑋 = 〈𝑥, 0〉) |
10 | 2, 3, 9 | 3bitri 297 | 1 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = 〈𝑥, 0〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 {csn 4630 〈cop 4636 × cxp 5686 (class class class)co 7430 0cc0 11152 ℤcz 12610 ×s cxps 17552 ℤringczring 21474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-1cn 11210 ax-addrcl 11213 ax-rnegex 11223 ax-cnre 11225 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-iota 6515 df-fv 6570 df-ov 7433 df-neg 11492 df-z 12611 |
This theorem is referenced by: pzriprnglem4 21512 pzriprnglem5 21513 pzriprnglem6 21514 pzriprnglem8 21516 |
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