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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 37025). (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 4163 | . . 3 ⊢ (𝑡 ∈ (𝑥 ∪ {𝑦}) ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ {𝑦})) | |
2 | velsn 4647 | . . . 4 ⊢ (𝑡 ∈ {𝑦} ↔ 𝑡 = 𝑦) | |
3 | 2 | orbi2i 912 | . . 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 37014 | 1 ⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∨ wo 847 ∀wal 1535 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 |
This theorem is referenced by: bj-adjg1 37026 |
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