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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 5035 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | nfrmo1 3374 | . . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfal 2341 | . 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
4 | 1, 3 | nfxfr 1852 | 1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1534 Ⅎwnf 1783 ∈ wcel 2113 ∃*wrmo 3144 Disj wdisj 5034 |
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 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1780 df-nf 1784 df-mo 2621 df-rmo 3149 df-disj 5035 |
This theorem is referenced by: disjabrex 30335 disjabrexf 30336 hasheuni 31348 ldgenpisyslem1 31426 measvunilem 31475 measvunilem0 31476 measvuni 31477 measinblem 31483 voliune 31492 volfiniune 31493 volmeas 31494 dstrvprob 31733 ismeannd 42756 |
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