Theorem dfeldisj3 35967

Description: Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)

Assertion

Ref	Expression
dfeldisj3	⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ∀𝑥 ∈ (𝑢 ∩ 𝑣)𝑢 = 𝑣)

Distinct variable group:   𝑢,𝐴,𝑣,𝑥

Proof of Theorem dfeldisj3

Step	Hyp	Ref	Expression
1		df-eldisj 35955	. . 3 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
2		relres 5882	. . . 4 ⊢ Rel (◡ E ↾ 𝐴)
3		dfdisjALTV3 35963	. . . 4 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ (∀𝑢∀𝑣∀𝑥((𝑢(◡ E ↾ 𝐴)𝑥 ∧ 𝑣(◡ E ↾ 𝐴)𝑥) → 𝑢 = 𝑣) ∧ Rel (◡ E ↾ 𝐴)))
4	2, 3	mpbiran2 708	. . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ∀𝑢∀𝑣∀𝑥((𝑢(◡ E ↾ 𝐴)𝑥 ∧ 𝑣(◡ E ↾ 𝐴)𝑥) → 𝑢 = 𝑣))
5		an4 654	. . . . . . 7 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝑣)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣)))
6		brcnvepres 35543	. . . . . . . . 9 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢(◡ E ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢)))
7	6	el2v 3501	. . . . . . . 8 ⊢ (𝑢(◡ E ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢))
8		brcnvepres 35543	. . . . . . . . 9 ⊢ ((𝑣 ∈ V ∧ 𝑥 ∈ V) → (𝑣(◡ E ↾ 𝐴)𝑥 ↔ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝑣)))
9	8	el2v 3501	. . . . . . . 8 ⊢ (𝑣(◡ E ↾ 𝐴)𝑥 ↔ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝑣))
10	7, 9	anbi12i 628	. . . . . . 7 ⊢ ((𝑢(◡ E ↾ 𝐴)𝑥 ∧ 𝑣(◡ E ↾ 𝐴)𝑥) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ 𝑢) ∧ (𝑣 ∈ 𝐴 ∧ 𝑥 ∈ 𝑣)))
11		elin 4169	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))
12	11	anbi2i 624	. . . . . . 7 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣)))
13	5, 10, 12	3bitr4i 305	. . . . . 6 ⊢ ((𝑢(◡ E ↾ 𝐴)𝑥 ∧ 𝑣(◡ E ↾ 𝐴)𝑥) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)))
14		df-3an 1085	. . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)))
15	13, 14	bitr4i 280	. . . . 5 ⊢ ((𝑢(◡ E ↾ 𝐴)𝑥 ∧ 𝑣(◡ E ↾ 𝐴)𝑥) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)))
16	15	imbi1i 352	. . . 4 ⊢ (((𝑢(◡ E ↾ 𝐴)𝑥 ∧ 𝑣(◡ E ↾ 𝐴)𝑥) → 𝑢 = 𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) → 𝑢 = 𝑣))
17	16	3albii 35524	. . 3 ⊢ (∀𝑢∀𝑣∀𝑥((𝑢(◡ E ↾ 𝐴)𝑥 ∧ 𝑣(◡ E ↾ 𝐴)𝑥) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) → 𝑢 = 𝑣))
18	1, 4, 17	3bitri 299	. 2 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢∀𝑣∀𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) → 𝑢 = 𝑣))
19		r3al 3202	. 2 ⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ∀𝑥 ∈ (𝑢 ∩ 𝑣)𝑢 = 𝑣 ↔ ∀𝑢∀𝑣∀𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴 ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) → 𝑢 = 𝑣))
20	18, 19	bitr4i 280	1 ⊢ ( ElDisj 𝐴 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ∀𝑥 ∈ (𝑢 ∩ 𝑣)𝑢 = 𝑣)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1535   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ∩ cin 3935   class class class wbr 5066   E cep 5464  ^◡ccnv 5554   ↾ cres 5557  Rel wrel 5560   Disj wdisjALTV 35502   ElDisj weldisj 35504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-eprel 5465  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-coss 35674  df-cnvrefrel 35780  df-disjALTV 35953  df-eldisj 35955

This theorem is referenced by: (None)