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| Mirrors > Home > MPE Home > Th. List > nfdju | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
| Ref | Expression |
|---|---|
| nfdju.1 | ⊢ Ⅎ𝑥𝐴 |
| nfdju.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| nfdju | ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-dju 9941 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
| 2 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥{∅} | |
| 3 | nfdju.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 2, 3 | nfxp 5718 | . . 3 ⊢ Ⅎ𝑥({∅} × 𝐴) |
| 5 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥{1o} | |
| 6 | nfdju.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 7 | 5, 6 | nfxp 5718 | . . 3 ⊢ Ⅎ𝑥({1o} × 𝐵) |
| 8 | 4, 7 | nfun 4170 | . 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
| 9 | 1, 8 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 ∪ cun 3949 ∅c0 4333 {csn 4626 × cxp 5683 1oc1o 8499 ⊔ cdju 9938 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-un 3956 df-opab 5206 df-xp 5691 df-dju 9941 |
| This theorem is referenced by: (None) |
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