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Mathbox for Peter Mazsa |
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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 38676 | . 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | |
2 | inecmo2 38312 | . . . 4 ⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ∧ Rel ◡ E ) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E )) | |
3 | relcnv 6134 | . . . . 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 38238 | . . . . . . . 8 ⊢ (𝑢 ∈ V → [𝑢]◡ E = 𝑢) | |
8 | 7 | elv 3493 | . . . . . . 7 ⊢ [𝑢]◡ E = 𝑢 |
9 | eccnvep 38238 | . . . . . . . 8 ⊢ (𝑣 ∈ V → [𝑣]◡ E = 𝑣) | |
10 | 9 | elv 3493 | . . . . . . 7 ⊢ [𝑣]◡ E = 𝑣 |
11 | 8, 10 | ineq12i 4239 | . . . . . 6 ⊢ ([𝑢]◡ E ∩ [𝑣]◡ E ) = (𝑢 ∩ 𝑣) |
12 | 11 | eqeq1i 2745 | . . . . 5 ⊢ (([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅ ↔ (𝑢 ∩ 𝑣) = ∅) |
13 | 12 | orbi2i 911 | . . . 4 ⊢ ((𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) |
14 | 13 | 2ralbii 3134 | . . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) |
15 | brcnvep 38221 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢)) | |
16 | 15 | elv 3493 | . . . . 5 ⊢ (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢) |
17 | 16 | rmobii 3396 | . . . 4 ⊢ (∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
18 | 17 | albii 1817 | . . 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 846 ∀wal 1535 = wceq 1537 ∀wral 3067 ∃*wrmo 3387 Vcvv 3488 ∩ cin 3975 ∅c0 4352 class class class wbr 5166 E cep 5598 ◡ccnv 5699 Rel wrel 5705 [cec 8761 ElDisj weldisj 38171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-id 5593 df-eprel 5599 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ec 8765 df-coss 38367 df-cnvrefrel 38483 df-disjALTV 38661 df-eldisj 38663 |
This theorem is referenced by: eqvreldisj2 38781 |
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