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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prclaxpr | Structured version Visualization version GIF version | ||
| Description: A class that is closed under the pairing operation models the Axiom of Pairing ax-pr 5430. Lemma II.2.4(4) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 29-Sep-2025.) |
| Ref | Expression |
|---|---|
| prclaxpr | ⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3483 | . . . . . 6 ⊢ 𝑤 ∈ V | |
| 2 | 1 | elpr 4648 | . . . . 5 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
| 3 | 2 | biimpri 228 | . . . 4 ⊢ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦}) |
| 4 | 3 | rgenw 3064 | . . 3 ⊢ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦}) |
| 5 | eleq2 2829 | . . . . . 6 ⊢ (𝑧 = {𝑥, 𝑦} → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
| 6 | 5 | imbi2d 340 | . . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} → (((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦}))) |
| 7 | 6 | ralbidv 3177 | . . . 4 ⊢ (𝑧 = {𝑥, 𝑦} → (∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦}))) |
| 8 | 7 | rspcev 3621 | . . 3 ⊢ (({𝑥, 𝑦} ∈ 𝑀 ∧ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})) → ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
| 9 | 4, 8 | mpan2 691 | . 2 ⊢ ({𝑥, 𝑦} ∈ 𝑀 → ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
| 10 | 9 | 2ralimi 3122 | 1 ⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3060 ∃wrex 3069 {cpr 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-v 3481 df-un 3955 df-sn 4625 df-pr 4627 |
| This theorem is referenced by: wfaxpr 44991 |
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