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Theorem elirr 4499
 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 4495, we could redefine Ord 𝐴 (df-iord 4326) to also require E Fr 𝐴 (df-frind 4292) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4500 (which under that definition would presumably not need ax-setind 4495 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 4500. To encourage ordirr 4500 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 ¬ 𝐴𝐴

Proof of Theorem elirr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neldifsnd 3690 . . . . . . . . 9 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝐴 ∈ (V ∖ {𝐴}))
2 simp1 982 . . . . . . . . . . 11 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → 𝐴𝐴)
3 eleq1 2220 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
4 eleq1 2220 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (𝑦 ∈ (V ∖ {𝐴}) ↔ 𝐴 ∈ (V ∖ {𝐴})))
53, 4imbi12d 233 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → ((𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) ↔ (𝐴𝑥𝐴 ∈ (V ∖ {𝐴}))))
65spcgv 2799 . . . . . . . . . . . . . 14 (𝐴𝑥 → (∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) → (𝐴𝑥𝐴 ∈ (V ∖ {𝐴}))))
76pm2.43b 52 . . . . . . . . . . . . 13 (∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) → (𝐴𝑥𝐴 ∈ (V ∖ {𝐴})))
873ad2ant2 1004 . . . . . . . . . . . 12 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → (𝐴𝑥𝐴 ∈ (V ∖ {𝐴})))
9 eleq2 2221 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
109imbi1d 230 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ((𝐴𝑥𝐴 ∈ (V ∖ {𝐴})) ↔ (𝐴𝐴𝐴 ∈ (V ∖ {𝐴}))))
11103ad2ant3 1005 . . . . . . . . . . . 12 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → ((𝐴𝑥𝐴 ∈ (V ∖ {𝐴})) ↔ (𝐴𝐴𝐴 ∈ (V ∖ {𝐴}))))
128, 11mpbid 146 . . . . . . . . . . 11 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → (𝐴𝐴𝐴 ∈ (V ∖ {𝐴})))
132, 12mpd 13 . . . . . . . . . 10 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) ∧ 𝑥 = 𝐴) → 𝐴 ∈ (V ∖ {𝐴}))
14133expia 1187 . . . . . . . . 9 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴}))) → (𝑥 = 𝐴𝐴 ∈ (V ∖ {𝐴})))
151, 14mtod 653 . . . . . . . 8 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴}))) → ¬ 𝑥 = 𝐴)
16 vex 2715 . . . . . . . . . 10 𝑥 ∈ V
17 eldif 3111 . . . . . . . . . 10 (𝑥 ∈ (V ∖ {𝐴}) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ {𝐴}))
1816, 17mpbiran 925 . . . . . . . . 9 (𝑥 ∈ (V ∖ {𝐴}) ↔ ¬ 𝑥 ∈ {𝐴})
19 velsn 3577 . . . . . . . . 9 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2018, 19xchbinx 672 . . . . . . . 8 (𝑥 ∈ (V ∖ {𝐴}) ↔ ¬ 𝑥 = 𝐴)
2115, 20sylibr 133 . . . . . . 7 ((𝐴𝐴 ∧ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴}))) → 𝑥 ∈ (V ∖ {𝐴}))
2221ex 114 . . . . . 6 (𝐴𝐴 → (∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
2322alrimiv 1854 . . . . 5 (𝐴𝐴 → ∀𝑥(∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
24 df-ral 2440 . . . . . . . 8 (∀𝑦𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ ∀𝑦(𝑦𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})))
25 clelsb3 2262 . . . . . . . . . 10 ([𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ 𝑦 ∈ (V ∖ {𝐴}))
2625imbi2i 225 . . . . . . . . 9 ((𝑦𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})) ↔ (𝑦𝑥𝑦 ∈ (V ∖ {𝐴})))
2726albii 1450 . . . . . . . 8 (∀𝑦(𝑦𝑥 → [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴})) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})))
2824, 27bitri 183 . . . . . . 7 (∀𝑦𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) ↔ ∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})))
2928imbi1i 237 . . . . . 6 ((∀𝑦𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})) ↔ (∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
3029albii 1450 . . . . 5 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})) ↔ ∀𝑥(∀𝑦(𝑦𝑥𝑦 ∈ (V ∖ {𝐴})) → 𝑥 ∈ (V ∖ {𝐴})))
3123, 30sylibr 133 . . . 4 (𝐴𝐴 → ∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})))
32 ax-setind 4495 . . . 4 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝑥 ∈ (V ∖ {𝐴}) → 𝑥 ∈ (V ∖ {𝐴})) → ∀𝑥 𝑥 ∈ (V ∖ {𝐴}))
3331, 32syl 14 . . 3 (𝐴𝐴 → ∀𝑥 𝑥 ∈ (V ∖ {𝐴}))
34 eleq1 2220 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ (V ∖ {𝐴}) ↔ 𝐴 ∈ (V ∖ {𝐴})))
3534spcgv 2799 . . 3 (𝐴𝐴 → (∀𝑥 𝑥 ∈ (V ∖ {𝐴}) → 𝐴 ∈ (V ∖ {𝐴})))
3633, 35mpd 13 . 2 (𝐴𝐴𝐴 ∈ (V ∖ {𝐴}))
37 neldifsnd 3690 . 2 (𝐴𝐴 → ¬ 𝐴 ∈ (V ∖ {𝐴}))
3836, 37pm2.65i 629 1 ¬ 𝐴𝐴
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963  ∀wal 1333   = wceq 1335  [wsb 1742   ∈ wcel 2128  ∀wral 2435  Vcvv 2712   ∖ cdif 3099  {csn 3560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4495 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-v 2714  df-dif 3104  df-sn 3566 This theorem is referenced by:  ordirr  4500  elirrv  4506  sucprcreg  4507  ordsoexmid  4520  onnmin  4526  ssnel  4527  ordtri2or2exmid  4529  reg3exmidlemwe  4537  nntri2  6438  nntri3  6441  nndceq  6443  nndcel  6444  phpelm  6808  fiunsnnn  6823  onunsnss  6858  snon0  6877
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