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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 5116 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | nfrmo1 3409 | . . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfal 2322 | . 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
4 | 1, 3 | nfxfr 1850 | 1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1535 Ⅎwnf 1780 ∈ wcel 2106 ∃*wrmo 3377 Disj wdisj 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-or 848 df-ex 1777 df-nf 1781 df-mo 2538 df-rmo 3378 df-disj 5116 |
This theorem is referenced by: disjabrex 32602 disjabrexf 32603 hasheuni 34066 ldgenpisyslem1 34144 measvunilem 34193 measvunilem0 34194 measvuni 34195 measinblem 34201 voliune 34210 volfiniune 34211 volmeas 34212 dstrvprob 34453 ismeannd 46423 |
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