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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 35987 | . 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) | |
2 | inecmo2 35644 | . . . 4 ⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ∧ Rel ◡ E ) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E )) | |
3 | relcnv 5960 | . . . . 5 ⊢ Rel ◡ E | |
4 | 3 | biantru 532 | . . . 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 532 | . . . 4 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ∧ Rel ◡ E )) |
6 | 2, 4, 5 | 3bitr4i 305 | . . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥) |
7 | eccnvep 35573 | . . . . . . . 8 ⊢ (𝑢 ∈ V → [𝑢]◡ E = 𝑢) | |
8 | 7 | elv 3496 | . . . . . . 7 ⊢ [𝑢]◡ E = 𝑢 |
9 | eccnvep 35573 | . . . . . . . 8 ⊢ (𝑣 ∈ V → [𝑣]◡ E = 𝑣) | |
10 | 9 | elv 3496 | . . . . . . 7 ⊢ [𝑣]◡ E = 𝑣 |
11 | 8, 10 | ineq12i 4180 | . . . . . 6 ⊢ ([𝑢]◡ E ∩ [𝑣]◡ E ) = (𝑢 ∩ 𝑣) |
12 | 11 | eqeq1i 2825 | . . . . 5 ⊢ (([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅ ↔ (𝑢 ∩ 𝑣) = ∅) |
13 | 12 | orbi2i 909 | . . . 4 ⊢ ((𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) |
14 | 13 | 2ralbii 3165 | . . 3 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]◡ E ∩ [𝑣]◡ E ) = ∅) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) |
15 | brcnvep 35560 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢)) | |
16 | 15 | elv 3496 | . . . . 5 ⊢ (𝑢◡ E 𝑥 ↔ 𝑥 ∈ 𝑢) |
17 | 16 | rmobii 3395 | . . . 4 ⊢ (∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
18 | 17 | albii 1819 | . . 3 ⊢ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢◡ E 𝑥 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
19 | 6, 14, 18 | 3bitr3i 303 | . 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑥 ∈ 𝑢) |
20 | 1, 19 | bitr4i 280 | 1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ (𝑢 ∩ 𝑣) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∨ wo 843 ∀wal 1534 = wceq 1536 ∀wral 3137 ∃*wrmo 3140 Vcvv 3491 ∩ cin 3928 ∅c0 4284 class class class wbr 5059 E cep 5457 ◡ccnv 5547 Rel wrel 5553 [cec 8280 ElDisj weldisj 35523 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8284 df-coss 35693 df-cnvrefrel 35799 df-disjALTV 35972 df-eldisj 35974 |
This theorem is referenced by: (None) |
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