Theorem elirr 4688

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 4684, we could redefine Ord 𝐴 (df-iord 4511) to also require E Fr 𝐴 (df-frind 4477) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4689 (which under that definition would presumably not need ax-setind 4684 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 4689. To encourage ordirr 4689 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 3845	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝐴 ∈ (V ∖ {𝐴}))
2		simp1 1028	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐴)
3		eleq1 2301	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
4		eleq1 2301	. . . . . . . . . . . . . . . 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 2912	. . . . . . . . . . . . . 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 1050	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝑥 → 𝐴 ∈ (V ∖ {𝐴})))
9		eleq2 2302	. . . . . . . . . . . . . 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 1051	. . . . . . . . . . . 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 1236	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → (𝑥 = 𝐴 → 𝐴 ∈ (V ∖ {𝐴})))
15	1, 14	mtod 673	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐴 ∧ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝑥 = 𝐴)
16		vex 2824	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		eldif 3229	. . . . . . . . . 10 ⊢ (𝑥 ∈ (V ∖ {𝐴}) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝐴}))
18	16, 17	mpbiran 953	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ {𝐴}) ↔ ¬ 𝑥 ∈ {𝐴})
19		velsn 3726	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
20	18, 19	xchbinx 693	. . . . . . . 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 1927	. . . . 5 ⊢ (𝐴 ∈ 𝐴 → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
24		df-ral 2533	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ ∀𝑦(𝑦 ∈ 𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})))
25		clelsb1 2343	. . . . . . . . . 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 1523	. . . . . . . 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 1523	. . . . 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 4684	. . . 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 2301	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (V ∖ {𝐴}) ↔ 𝐴 ∈ (V ∖ {𝐴})))
35	34	spcgv 2912	. . 3 ⊢ (𝐴 ∈ 𝐴 → (∀𝑥 𝑥 ∈ (V ∖ {𝐴}) → 𝐴 ∈ (V ∖ {𝐴})))
36	33, 35	mpd 13	. 2 ⊢ (𝐴 ∈ 𝐴 → 𝐴 ∈ (V ∖ {𝐴}))
37		neldifsnd 3845	. 2 ⊢ (𝐴 ∈ 𝐴 → ¬ 𝐴 ∈ (V ∖ {𝐴}))
38	36, 37	pm2.65i 648	1 ⊢ ¬ 𝐴 ∈ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∀wal 1400   = wceq 1402  [wsb 1815   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ∖ cdif 3217  {csn 3709

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-sn 3715

This theorem is used by:  ordirr  4689  elirrv  4695  sucprcreg  4696  ordsoexmid  4709  onnmin  4715  ssnel  4716  ordtri2or2exmid  4718  reg3exmidlemwe  4726  nntri2  6767  nntri3  6770  nndceq  6772  nndcel  6773  phpelm  7168  fiunsnnn  7185  onunsnss  7224  snon0  7249