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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 5113 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | nfrmo1 3408 | . . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfal 2317 | . 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
4 | 1, 3 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1540 Ⅎwnf 1786 ∈ wcel 2107 ∃*wrmo 3376 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-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-or 847 df-ex 1783 df-nf 1787 df-mo 2535 df-rmo 3377 df-disj 5113 |
This theorem is referenced by: disjabrex 31791 disjabrexf 31792 hasheuni 33021 ldgenpisyslem1 33099 measvunilem 33148 measvunilem0 33149 measvuni 33150 measinblem 33156 voliune 33165 volfiniune 33166 volmeas 33167 dstrvprob 33408 ismeannd 45118 |
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