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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-adjg1 | Structured version Visualization version GIF version | ||
| Description: Existence of the result of the adjunction (generalized only in the first term since this suffices for current applications). (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-adjg1 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uneq1 4161 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∪ {𝑥}) = (𝐴 ∪ {𝑥})) | |
| 2 | 1 | eleq1d 2826 | . 2 ⊢ (𝑦 = 𝐴 → ((𝑦 ∪ {𝑥}) ∈ V ↔ (𝐴 ∪ {𝑥}) ∈ V)) | 
| 3 | ax-bj-adj 37043 | . . . . 5 ⊢ ∀𝑦∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥)) | |
| 4 | 3 | spi 2184 | . . . 4 ⊢ ∀𝑥∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥)) | 
| 5 | 4 | spi 2184 | . . 3 ⊢ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥)) | 
| 6 | bj-axadj 37042 | . . 3 ⊢ ((𝑦 ∪ {𝑥}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑦 ∨ 𝑡 = 𝑥))) | |
| 7 | 5, 6 | mpbir 231 | . 2 ⊢ (𝑦 ∪ {𝑥}) ∈ V | 
| 8 | 2, 7 | vtoclg 3554 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 848 ∀wal 1538 = wceq 1540 ∃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-12 2177 ax-ext 2708 ax-bj-adj 37043 | 
| 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-snfromadj 37045 bj-prfromadj 37046 | 
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