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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfeldisj5 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| dfeldisj5 | ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfeldisj4 38722 | . 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | |
| 2 | inecmo2 38358 | . . . 4 ⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ∧ Rel ◡ E ) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E )) | |
| 3 | relcnv 6121 | . . . . 5 ⊢ Rel ◡ E | |
| 4 | 3 | biantru 529 | . . . 4 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ∧ Rel ◡ E )) | 
| 5 | 3 | biantru 529 | . . . 4 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E )) | 
| 6 | 2, 4, 5 | 3bitr4i 303 | . . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥) | 
| 7 | eccnvep 38284 | . . . . . . . 8 ⊢ (𝑢 ∈ V → [𝑢]◡ E = 𝑢) | |
| 8 | 7 | elv 3484 | . . . . . . 7 ⊢ [𝑢]◡ E = 𝑢 | 
| 9 | eccnvep 38284 | . . . . . . . 8 ⊢ (𝑣 ∈ V → [𝑣]◡ E = 𝑣) | |
| 10 | 9 | elv 3484 | . . . . . . 7 ⊢ [𝑣]◡ E = 𝑣 | 
| 11 | 8, 10 | ineq12i 4217 | . . . . . 6 ⊢ ([𝑢]◡ E ∩ [𝑣]◡ E ) = (𝑢 ∩ 𝑣) | 
| 12 | 11 | eqeq1i 2741 | . . . . 5 ⊢ (([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅ ↔ (𝑢 ∩ 𝑣) = ∅) | 
| 13 | 12 | orbi2i 912 | . . . 4 ⊢ ((𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) | 
| 14 | 13 | 2ralbii 3127 | . . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) | 
| 15 | brcnvep 38267 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢)) | |
| 16 | 15 | elv 3484 | . . . . 5 ⊢ (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢) | 
| 17 | 16 | rmobii 3387 | . . . 4 ⊢ (∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | 
| 18 | 17 | albii 1818 | . . 3 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | 
| 19 | 6, 14, 18 | 3bitr3i 301 | . 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | 
| 20 | 1, 19 | bitr4i 278 | 1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∨ wo 847 ∀wal 1537 = wceq 1539 ∀wral 3060 ∃*wrmo 3378 Vcvv 3479 ∩ cin 3949 ∅c0 4332 class class class wbr 5142 E cep 5582 ◡ccnv 5683 Rel wrel 5689 [cec 8744 ElDisj weldisj 38219 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-eprel 5583 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ec 8748 df-coss 38413 df-cnvrefrel 38529 df-disjALTV 38707 df-eldisj 38709 | 
| This theorem is referenced by: eqvreldisj2 38827 | 
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