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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 9323 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | nfcv 2976 | . . . 4 ⊢ Ⅎ𝑥{∅} | |
3 | nfdju.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfxp 5581 | . . 3 ⊢ Ⅎ𝑥({∅} × 𝐴) |
5 | nfcv 2976 | . . . 4 ⊢ Ⅎ𝑥{1o} | |
6 | nfdju.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfxp 5581 | . . 3 ⊢ Ⅎ𝑥({1o} × 𝐵) |
8 | 4, 7 | nfun 4134 | . 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
9 | 1, 8 | nfcxfr 2974 | 1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2960 ∪ cun 3927 ∅c0 4284 {csn 4560 × cxp 5546 1oc1o 8088 ⊔ cdju 9320 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-un 3934 df-opab 5122 df-xp 5554 df-dju 9323 |
This theorem is referenced by: (None) |
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