Theorem elirr 4577

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 4573, we could redefine Ord 𝐴 (df-iord 4401) to also require E Fr 𝐴 (df-frind 4367) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4578 (which under that definition would presumably not need ax-setind 4573 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 4578. To encourage ordirr 4578 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 3753	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝐴 ∈ (V ∖ {𝐴}))
2		simp1 999	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
3		eleq1 2259	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
4		eleq1 2259	. . . . . . . . . . . . . . . 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 2851	. . . . . . . . . . . . . 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 1021	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴})))
9		eleq2 2260	. . . . . . . . . . . . . 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 1022	. . . . . . . . . . . 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 1207	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → (𝑥 = 𝐴 → 𝐴 ∈ (V ∖ {𝐴})))
15	1, 14	mtod 664	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝑥 = 𝐴)
16		vex 2766	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		eldif 3166	. . . . . . . . . 10 ⊢ (𝑥 ∈ (V ∖ {𝐴}) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝐴}))
18	16, 17	mpbiran 942	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ {𝐴}) ↔ ¬ 𝑥 ∈ {𝐴})
19		velsn 3639	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
20	18, 19	xchbinx 683	. . . . . . . 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 1888	. . . . 5 ⊢ (𝐴 ∈ 𝐴 → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
24		df-ral 2480	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ ∀𝑦(𝑦 ∈ 𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})))
25		clelsb1 2301	. . . . . . . . . 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 1484	. . . . . . . 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 1484	. . . . 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 4573	. . . 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 2259	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (V ∖ {𝐴}) ↔ 𝐴 ∈ (V ∖ {𝐴})))
35	34	spcgv 2851	. . 3 ⊢ (𝐴 ∈ 𝐴 → (∀𝑥 𝑥 ∈ (V ∖ {𝐴}) → 𝐴 ∈ (V ∖ {𝐴})))
36	33, 35	mpd 13	. 2 ⊢ (𝐴 ∈ 𝐴 → 𝐴 ∈ (V ∖ {𝐴}))
37		neldifsnd 3753	. 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ (V ∖ {𝐴}))
38	36, 37	pm2.65i 640	1 ⊢ ¬ 𝐴 ∈ 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980  ∀wal 1362   = wceq 1364  [wsb 1776   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ∖ cdif 3154  {csn 3622

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-v 2765  df-dif 3159  df-sn 3628

This theorem is referenced by:  ordirr  4578  elirrv  4584  sucprcreg  4585  ordsoexmid  4598  onnmin  4604  ssnel  4605  ordtri2or2exmid  4607  reg3exmidlemwe  4615  nntri2  6552  nntri3  6555  nndceq  6557  nndcel  6558  phpelm  6927  fiunsnnn  6942  onunsnss  6978  snon0  7001