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Mirrors > Home > MPE Home > Th. List > nfdisj | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for disjoint collection. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfdisjw 5007 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfdisj.1 | ⊢ Ⅎ𝑦𝐴 |
nfdisj.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfdisj | ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisj2 4997 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
2 | nftru 1806 | . . . . 5 ⊢ Ⅎ𝑥⊤ | |
3 | nfcvf 2981 | . . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
4 | nfdisj.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
5 | 4 | a1i 11 | . . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝐴) |
6 | 3, 5 | nfeld 2966 | . . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝐴) |
7 | nfdisj.2 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐵 | |
8 | 7 | nfcri 2943 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧 ∈ 𝐵) |
10 | 6, 9 | nfand 1898 | . . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
11 | 10 | adantl 485 | . . . . 5 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
12 | 2, 11 | nfmod2 2617 | . . . 4 ⊢ (⊤ → Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
13 | 12 | mptru 1545 | . . 3 ⊢ Ⅎ𝑦∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
14 | 13 | nfal 2331 | . 2 ⊢ Ⅎ𝑦∀𝑧∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
15 | 1, 14 | nfxfr 1854 | 1 ⊢ Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∀wal 1536 ⊤wtru 1539 Ⅎwnf 1785 ∈ wcel 2111 ∃*wmo 2596 Ⅎwnfc 2936 Disj wdisj 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-mo 2598 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rmo 3114 df-disj 4996 |
This theorem is referenced by: (None) |
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