Theorem antid 5930

Description: The antisymmetry property. (Contributed by SF, 18-Mar-2015.)

Hypotheses

Ref	Expression
antid.1	⊢ (φ → R Antisym A)
antid.2	⊢ (φ → X ∈ A)
antid.3	⊢ (φ → Y ∈ A)
antid.4	⊢ (φ → XRY)
antid.5	⊢ (φ → YRX)

Assertion

Ref	Expression
antid	⊢ (φ → X = Y)

Proof of Theorem antid

Dummy variables a r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		antid.4	. 2 ⊢ (φ → XRY)
2		antid.5	. 2 ⊢ (φ → YRX)
3		antid.1	. . . 4 ⊢ (φ → R Antisym A)
4		brex 4690	. . . . . 6 ⊢ (R Antisym A → (R ∈ V ∧ A ∈ V))
5		breq 4642	. . . . . . . . . 10 ⊢ (r = R → (xry ↔ xRy))
6		breq 4642	. . . . . . . . . 10 ⊢ (r = R → (yrx ↔ yRx))
7	5, 6	anbi12d 691	. . . . . . . . 9 ⊢ (r = R → ((xry ∧ yrx) ↔ (xRy ∧ yRx)))
8	7	imbi1d 308	. . . . . . . 8 ⊢ (r = R → (((xry ∧ yrx) → x = y) ↔ ((xRy ∧ yRx) → x = y)))
9	8	2ralbidv 2657	. . . . . . 7 ⊢ (r = R → (∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y) ↔ ∀x ∈ a ∀y ∈ a ((xRy ∧ yRx) → x = y)))
10		raleq 2808	. . . . . . . 8 ⊢ (a = A → (∀y ∈ a ((xRy ∧ yRx) → x = y) ↔ ∀y ∈ A ((xRy ∧ yRx) → x = y)))
11	10	raleqbi1dv 2816	. . . . . . 7 ⊢ (a = A → (∀x ∈ a ∀y ∈ a ((xRy ∧ yRx) → x = y) ↔ ∀x ∈ A ∀y ∈ A ((xRy ∧ yRx) → x = y)))
12		df-antisym 5902	. . . . . . 7 ⊢ Antisym = {⟨r, a⟩ ∣ ∀x ∈ a ∀y ∈ a ((xry ∧ yrx) → x = y)}
13	9, 11, 12	brabg 4707	. . . . . 6 ⊢ ((R ∈ V ∧ A ∈ V) → (R Antisym A ↔ ∀x ∈ A ∀y ∈ A ((xRy ∧ yRx) → x = y)))
14	4, 13	syl 15	. . . . 5 ⊢ (R Antisym A → (R Antisym A ↔ ∀x ∈ A ∀y ∈ A ((xRy ∧ yRx) → x = y)))
15	14	ibi 232	. . . 4 ⊢ (R Antisym A → ∀x ∈ A ∀y ∈ A ((xRy ∧ yRx) → x = y))
16	3, 15	syl 15	. . 3 ⊢ (φ → ∀x ∈ A ∀y ∈ A ((xRy ∧ yRx) → x = y))
17		antid.2	. . . 4 ⊢ (φ → X ∈ A)
18		antid.3	. . . 4 ⊢ (φ → Y ∈ A)
19		breq1 4643	. . . . . . 7 ⊢ (x = X → (xRy ↔ XRy))
20		breq2 4644	. . . . . . 7 ⊢ (x = X → (yRx ↔ yRX))
21	19, 20	anbi12d 691	. . . . . 6 ⊢ (x = X → ((xRy ∧ yRx) ↔ (XRy ∧ yRX)))
22		eqeq1 2359	. . . . . 6 ⊢ (x = X → (x = y ↔ X = y))
23	21, 22	imbi12d 311	. . . . 5 ⊢ (x = X → (((xRy ∧ yRx) → x = y) ↔ ((XRy ∧ yRX) → X = y)))
24		breq2 4644	. . . . . . 7 ⊢ (y = Y → (XRy ↔ XRY))
25		breq1 4643	. . . . . . 7 ⊢ (y = Y → (yRX ↔ YRX))
26	24, 25	anbi12d 691	. . . . . 6 ⊢ (y = Y → ((XRy ∧ yRX) ↔ (XRY ∧ YRX)))
27		eqeq2 2362	. . . . . 6 ⊢ (y = Y → (X = y ↔ X = Y))
28	26, 27	imbi12d 311	. . . . 5 ⊢ (y = Y → (((XRy ∧ yRX) → X = y) ↔ ((XRY ∧ YRX) → X = Y)))
29	23, 28	rspc2v 2962	. . . 4 ⊢ ((X ∈ A ∧ Y ∈ A) → (∀x ∈ A ∀y ∈ A ((xRy ∧ yRx) → x = y) → ((XRY ∧ YRX) → X = Y)))
30	17, 18, 29	syl2anc 642	. . 3 ⊢ (φ → (∀x ∈ A ∀y ∈ A ((xRy ∧ yRx) → x = y) → ((XRY ∧ YRX) → X = Y)))
31	16, 30	mpd 14	. 2 ⊢ (φ → ((XRY ∧ YRX) → X = Y))
32	1, 2, 31	mp2and 660	1 ⊢ (φ → X = Y)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   class class class wbr 4640   Antisym cantisym 5891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-antisym 5902

This theorem is referenced by:  nchoicelem8  6297  nchoicelem19  6308