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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 5123 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 14-Nov-2016.) Avoid ax-13 2377. (Revised by GG, 26-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| nfdisjw.1 | ⊢ Ⅎ𝑦𝐴 | 
| nfdisjw.2 | ⊢ Ⅎ𝑦𝐵 | 
| Ref | Expression | 
|---|---|
| nfdisjw | ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfdisj2 5112 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
| 2 | nftru 1804 | . . . . 5 ⊢ Ⅎ𝑥⊤ | |
| 3 | nfcvd 2906 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝑥) | |
| 4 | nfdisjw.1 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐴 | |
| 5 | 4 | a1i 11 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦𝐴) | 
| 6 | 3, 5 | nfeld 2917 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴) | 
| 7 | nfdisjw.2 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐵 | |
| 8 | 7 | nfcri 2897 | . . . . . . 7 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 | 
| 9 | 8 | a1i 11 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦 𝑧 ∈ 𝐵) | 
| 10 | 6, 9 | nfand 1897 | . . . . 5 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | 
| 11 | 2, 10 | nfmodv 2559 | . . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | 
| 12 | 11 | mptru 1547 | . . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) | 
| 13 | 12 | nfal 2323 | . 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) | 
| 14 | 1, 13 | nfxfr 1853 | 1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∀wal 1538 ⊤wtru 1541 Ⅎwnf 1783 ∈ wcel 2108 ∃*wmo 2538 Ⅎwnfc 2890 Disj wdisj 5110 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rmo 3380 df-disj 5111 | 
| This theorem is referenced by: disjxiun 5140 | 
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