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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 5111 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
| 2 | nfrmo1 3411 | . . 3 ⊢ Ⅎ𝑥∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
| 3 | 2 | nfal 2323 | . 2 ⊢ Ⅎ𝑥∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 |
| 4 | 1, 3 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ∀wal 1538 Ⅎwnf 1783 ∈ wcel 2108 ∃*wrmo 3379 Disj wdisj 5110 |
| 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 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 df-mo 2540 df-rmo 3380 df-disj 5111 |
| This theorem is referenced by: disjabrex 32595 disjabrexf 32596 hasheuni 34086 ldgenpisyslem1 34164 measvunilem 34213 measvunilem0 34214 measvuni 34215 measinblem 34221 voliune 34230 volfiniune 34231 volmeas 34232 dstrvprob 34474 ismeannd 46482 |
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