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Mathbox for Peter Mazsa |
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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 37219 | . 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | |
2 | relres 5970 | . . 3 ⊢ Rel (◡ E ↾ 𝐴) | |
3 | dfdisjALTV4 37228 | . . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ (∀𝑥∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥 ∧ Rel (◡ E ↾ 𝐴))) | |
4 | 2, 3 | mpbiran2 709 | . 2 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ∀𝑥∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥) |
5 | brcnvepres 36777 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢(◡ E ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢))) | |
6 | 5 | el2v 3455 | . . . . 5 ⊢ (𝑢(◡ E ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) |
7 | 6 | mobii 2543 | . . . 4 ⊢ (∃*𝑢 𝑢(◡ E ↾ 𝐴)𝑥 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)) |
8 | df-rmo 3352 | . . . 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 1822 | . 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 205 ∧ wa 397 ∀wal 1540 ∈ wcel 2107 ∃*wmo 2533 ∃*wrmo 3351 Vcvv 3447 class class class wbr 5109 E cep 5540 ◡ccnv 5636 ↾ cres 5639 Rel wrel 5642 Disj wdisjALTV 36718 ElDisj weldisj 36720 |
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-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-id 5535 df-eprel 5541 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-coss 36923 df-cnvrefrel 37039 df-disjALTV 37217 df-eldisj 37219 |
This theorem is referenced by: dfeldisj5 37233 |
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