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Mirrors > Home > MPE Home > Th. List > nfdisj1 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
nfdisj1 | ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-disj 4996 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | nfrmo1 3274 | . . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfal 2325 | . 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
4 | 1, 3 | nfxfr 1859 | 1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1540 Ⅎwnf 1790 ∈ wcel 2114 ∃*wrmo 3056 Disj wdisj 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-or 847 df-ex 1787 df-nf 1791 df-mo 2540 df-rmo 3061 df-disj 4996 |
This theorem is referenced by: disjabrex 30495 disjabrexf 30496 hasheuni 31623 ldgenpisyslem1 31701 measvunilem 31750 measvunilem0 31751 measvuni 31752 measinblem 31758 voliune 31767 volfiniune 31768 volmeas 31769 dstrvprob 32008 ismeannd 43547 |
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