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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-axadj | Structured version Visualization version GIF version | ||
| Description: Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 37043). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-axadj | ⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elun 4153 | . . 3 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦})) | |
| 2 | velsn 4642 | . . . 4 ⊢ (𝑡 ∈ {𝑦} ↔ 𝑡 = 𝑦) | |
| 3 | 2 | orbi2i 913 | . . 3 ⊢ ((𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)) | 
| 4 | 1, 3 | bitri 275 | . 2 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)) | 
| 5 | 4 | bj-clex 37032 | 1 ⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 848 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 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 2708 | 
| 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 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 | 
| This theorem is referenced by: bj-adjg1 37044 | 
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