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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 9924 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥{∅} | |
3 | nfdju.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfxp 5711 | . . 3 ⊢ Ⅎ𝑥({∅} × 𝐴) |
5 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥{1o} | |
6 | nfdju.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfxp 5711 | . . 3 ⊢ Ⅎ𝑥({1o} × 𝐵) |
8 | 4, 7 | nfun 4164 | . 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
9 | 1, 8 | nfcxfr 2897 | 1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2879 ∪ cun 3945 ∅c0 4323 {csn 4629 × cxp 5676 1oc1o 8479 ⊔ cdju 9921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-v 3473 df-un 3952 df-opab 5211 df-xp 5684 df-dju 9924 |
This theorem is referenced by: (None) |
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