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Mirrors > Home > MPE Home > Th. List > nfdisjw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5126 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2370. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
nfdisjw.1 | ⊢ Ⅎ𝑦𝐴 |
nfdisjw.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfdisjw | ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisj2 5115 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nftru 1805 | . . . . 5 ⊢ Ⅎ𝑥⊤ | |
3 | nfcvd 2903 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝑥) | |
4 | nfdisjw.1 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐴 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴) |
6 | 3, 5 | nfeld 2913 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴) |
7 | nfdisjw.2 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐵 | |
8 | 7 | nfcri 2889 | . . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵) |
10 | 6, 9 | nfand 1899 | . . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
11 | 2, 10 | nfmodv 2552 | . . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
12 | 11 | mptru 1547 | . . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
13 | 12 | nfal 2315 | . 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
14 | 1, 13 | nfxfr 1854 | 1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∀wal 1538 ⊤wtru 1541 Ⅎwnf 1784 ∈ wcel 2105 ∃*wmo 2531 Ⅎwnfc 2882 Disj wdisj 5113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-mo 2533 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rmo 3375 df-disj 5114 |
This theorem is referenced by: disjxiun 5145 |
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