| Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfeldisj4 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) |
| Ref | Expression |
|---|---|
| dfeldisj4 | ⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-eldisj 38708 | . 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | |
| 2 | relres 6023 | . . 3 ⊢ Rel (◡ E ↾ 𝐴) | |
| 3 | dfdisjALTV4 38717 | . . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ (∀𝑥∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥 ∧ Rel (◡ E ↾ 𝐴))) | |
| 4 | 2, 3 | mpbiran2 710 | . 2 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ∀𝑥∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥) |
| 5 | brcnvepres 38268 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢(◡ E ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢))) | |
| 6 | 5 | el2v 3487 | . . . . 5 ⊢ (𝑢(◡ E ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) |
| 7 | 6 | mobii 2548 | . . . 4 ⊢ (∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) |
| 8 | df-rmo 3380 | . . . 4 ⊢ (∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) | |
| 9 | 7, 8 | bitr4i 278 | . . 3 ⊢ (∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
| 10 | 9 | albii 1819 | . 2 ⊢ (∀𝑥∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
| 11 | 1, 4, 10 | 3bitri 297 | 1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 ∃*wmo 2538 ∃*wrmo 3379 Vcvv 3480 class class class wbr 5143 E cep 5583 ◡ccnv 5684 ↾ cres 5687 Rel wrel 5690 Disj wdisjALTV 38216 ElDisj weldisj 38218 |
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-coss 38412 df-cnvrefrel 38528 df-disjALTV 38706 df-eldisj 38708 |
| This theorem is referenced by: dfeldisj5 38722 |
| Copyright terms: Public domain | W3C validator |