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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 9893 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
2 | nfcv 2895 | . . . 4 ⊢ Ⅎ𝑥{∅} | |
3 | nfdju.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nfxp 5700 | . . 3 ⊢ Ⅎ𝑥({∅} × 𝐴) |
5 | nfcv 2895 | . . . 4 ⊢ Ⅎ𝑥{1o} | |
6 | nfdju.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfxp 5700 | . . 3 ⊢ Ⅎ𝑥({1o} × 𝐵) |
8 | 4, 7 | nfun 4158 | . 2 ⊢ Ⅎ𝑥(({∅} × 𝐴) ∪ ({1o} × 𝐵)) |
9 | 1, 8 | nfcxfr 2893 | 1 ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2875 ∪ cun 3939 ∅c0 4315 {csn 4621 × cxp 5665 1oc1o 8455 ⊔ cdju 9890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-un 3946 df-opab 5202 df-xp 5673 df-dju 9893 |
This theorem is referenced by: (None) |
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