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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 5125 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2372. (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 5114 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nftru 1807 | . . . . 5 ⊢ Ⅎ𝑥⊤ | |
3 | nfcvd 2905 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝑥) | |
4 | nfdisjw.1 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐴 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴) |
6 | 3, 5 | nfeld 2915 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴) |
7 | nfdisjw.2 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐵 | |
8 | 7 | nfcri 2891 | . . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵) |
10 | 6, 9 | nfand 1901 | . . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
11 | 2, 10 | nfmodv 2554 | . . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
12 | 11 | mptru 1549 | . . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
13 | 12 | nfal 2317 | . 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
14 | 1, 13 | nfxfr 1856 | 1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∀wal 1540 ⊤wtru 1543 Ⅎwnf 1786 ∈ wcel 2107 ∃*wmo 2533 Ⅎwnfc 2884 Disj wdisj 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-mo 2535 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rmo 3377 df-disj 5113 |
This theorem is referenced by: disjxiun 5144 |
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