Theorem elirr 4665

Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22.

The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4661, we could redefine Ord 𝐴 (df-iord 4489) to also require E Fr 𝐴 (df-frind 4455) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4666 (which under that definition would presumably not need ax-setind 4661 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4666. To encourage ordirr 4666 when possible, we mark this theorem as discouraged.

(Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
elirr	⊢ ¬ 𝐴 ∈ 𝐴

Proof of Theorem elirr

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

Step	Hyp	Ref	Expression
1		neldifsnd 3826	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝐴 ∈ (V ∖ {𝐴}))
2		simp1 1024	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
3		eleq1 2297	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
4		eleq1 2297	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (V ∖ {𝐴}) ↔ 𝐴 ∈ (V ∖ {𝐴})))
5	3, 4	imbi12d 234	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ↔ (𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴}))))
6	5	spcgv 2906	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑥 → (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → (𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴}))))
7	6	pm2.43b 52	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → (𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴})))
8	7	3ad2ant2 1046	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴})))
9		eleq2 2298	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐴))
10	9	imbi1d 231	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → ((𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴})) ↔ (𝐴 ∈ 𝐴 → 𝐴 ∈ (V ∖ {𝐴}))))
11	10	3ad2ant3 1047	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → ((𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴})) ↔ (𝐴 ∈ 𝐴 → 𝐴 ∈ (V ∖ {𝐴}))))
12	8, 11	mpbid 147	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐴 → 𝐴 ∈ (V ∖ {𝐴})))
13	2, 12	mpd 13	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → 𝐴 ∈ (V ∖ {𝐴}))
14	13	3expia 1232	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → (𝑥 = 𝐴 → 𝐴 ∈ (V ∖ {𝐴})))
15	1, 14	mtod 669	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝑥 = 𝐴)
16		vex 2818	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		eldif 3222	. . . . . . . . . 10 ⊢ (𝑥 ∈ (V ∖ {𝐴}) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝐴}))
18	16, 17	mpbiran 949	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ {𝐴}) ↔ ¬ 𝑥 ∈ {𝐴})
19		velsn 3708	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
20	18, 19	xchbinx 689	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ {𝐴}) ↔ ¬ 𝑥 = 𝐴)
21	15, 20	sylibr 134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → 𝑥 ∈ (V ∖ {𝐴}))
22	21	ex 115	. . . . . 6 ⊢ (𝐴 ∈ 𝐴 → (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
23	22	alrimiv 1923	. . . . 5 ⊢ (𝐴 ∈ 𝐴 → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
24		df-ral 2527	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ ∀𝑦(𝑦 ∈ 𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})))
25		clelsb1 2339	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ 𝑦 ∈ (V ∖ {𝐴}))
26	25	imbi2i 226	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})) ↔ (𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})))
27	26	albii 1519	. . . . . . . 8 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})))
28	24, 27	bitri 184	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})))
29	28	imbi1i 238	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})) ↔ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
30	29	albii 1519	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
31	23, 30	sylibr 134	. . . 4 ⊢ (𝐴 ∈ 𝐴 → ∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})))
32		ax-setind 4661	. . . 4 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})) → ∀𝑥 𝑥 ∈ (V ∖ {𝐴}))
33	31, 32	syl 14	. . 3 ⊢ (𝐴 ∈ 𝐴 → ∀𝑥 𝑥 ∈ (V ∖ {𝐴}))
34		eleq1 2297	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (V ∖ {𝐴}) ↔ 𝐴 ∈ (V ∖ {𝐴})))
35	34	spcgv 2906	. . 3 ⊢ (𝐴 ∈ 𝐴 → (∀𝑥 𝑥 ∈ (V ∖ {𝐴}) → 𝐴 ∈ (V ∖ {𝐴})))
36	33, 35	mpd 13	. 2 ⊢ (𝐴 ∈ 𝐴 → 𝐴 ∈ (V ∖ {𝐴}))
37		neldifsnd 3826	. 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ (V ∖ {𝐴}))
38	36, 37	pm2.65i 644	1 ⊢ ¬ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005  ∀wal 1396   = wceq 1398  [wsb 1811   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ∖ cdif 3210  {csn 3691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-v 2817  df-dif 3215  df-sn 3697

This theorem is referenced by:  ordirr  4666  elirrv  4672  sucprcreg  4673  ordsoexmid  4686  onnmin  4692  ssnel  4693  ordtri2or2exmid  4695  reg3exmidlemwe  4703  nntri2  6729  nntri3  6732  nndceq  6734  nndcel  6735  phpelm  7123  fiunsnnn  7140  onunsnss  7179  snon0  7204