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Theorem bj-nnelirr 13140
Description: A natural number does not belong to itself. Version of elirr 4451 for natural numbers, which does not require ax-setind 4447. (Contributed by BJ, 24-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nnelirr (𝐴 ∈ ω → ¬ 𝐴𝐴)

Proof of Theorem bj-nnelirr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3362 . 2 ¬ ∅ ∈ ∅
2 df-suc 4288 . . . . . 6 suc 𝑦 = (𝑦 ∪ {𝑦})
32eleq2i 2204 . . . . 5 (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4 elun 3212 . . . . . 6 (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5 bj-nntrans 13138 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦 → suc 𝑦𝑦))
6 sucssel 4341 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
75, 6syld 45 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
8 vex 2684 . . . . . . . . . 10 𝑦 ∈ V
98sucid 4334 . . . . . . . . 9 𝑦 ∈ suc 𝑦
10 elsni 3540 . . . . . . . . 9 (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
119, 10eleqtrid 2226 . . . . . . . 8 (suc 𝑦 ∈ {𝑦} → 𝑦𝑦)
1211a1i 9 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦𝑦))
137, 12jaod 706 . . . . . 6 (𝑦 ∈ ω → ((suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}) → 𝑦𝑦))
144, 13syl5bi 151 . . . . 5 (𝑦 ∈ ω → (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) → 𝑦𝑦))
153, 14syl5bi 151 . . . 4 (𝑦 ∈ ω → (suc 𝑦 ∈ suc 𝑦𝑦𝑦))
1615con3d 620 . . 3 (𝑦 ∈ ω → (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
1716rgen 2483 . 2 𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18 ax-bdel 13008 . . . 4 BOUNDED 𝑥𝑥
1918ax-bdn 13004 . . 3 BOUNDED ¬ 𝑥𝑥
20 nfv 1508 . . 3 𝑥 ¬ ∅ ∈ ∅
21 nfv 1508 . . 3 𝑥 ¬ 𝑦𝑦
22 nfv 1508 . . 3 𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23 eleq1 2200 . . . . . 6 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ 𝑥))
24 eleq2 2201 . . . . . 6 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2523, 24bitrd 187 . . . . 5 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ ∅))
2625notbid 656 . . . 4 (𝑥 = ∅ → (¬ 𝑥𝑥 ↔ ¬ ∅ ∈ ∅))
2726biimprd 157 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥𝑥))
28 elequ1 1690 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
29 elequ2 1691 . . . . . 6 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
3028, 29bitrd 187 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
3130notbid 656 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
3231biimpd 143 . . 3 (𝑥 = 𝑦 → (¬ 𝑥𝑥 → ¬ 𝑦𝑦))
33 eleq1 2200 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦𝑥))
34 eleq2 2201 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3533, 34bitrd 187 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3635notbid 656 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
3736biimprd 157 . . 3 (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥𝑥))
38 nfcv 2279 . . 3 𝑥𝐴
39 nfv 1508 . . 3 𝑥 ¬ 𝐴𝐴
40 eleq1 2200 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
41 eleq2 2201 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
4240, 41bitrd 187 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
4342notbid 656 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4443biimpd 143 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 → ¬ 𝐴𝐴))
4519, 20, 21, 22, 27, 32, 37, 38, 39, 44bj-bdfindisg 13135 . 2 ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴𝐴))
461, 17, 45mp2an 422 1 (𝐴 ∈ ω → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 697   = wceq 1331  wcel 1480  wral 2414  cun 3064  wss 3066  c0 3358  {csn 3522  suc csuc 4282  ωcom 4499
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049  ax-pr 4126  ax-un 4350  ax-bd0 13000  ax-bdor 13003  ax-bdn 13004  ax-bdal 13005  ax-bdex 13006  ax-bdeq 13007  ax-bdel 13008  ax-bdsb 13009  ax-bdsep 13071  ax-infvn 13128
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-sn 3528  df-pr 3529  df-uni 3732  df-int 3767  df-suc 4288  df-iom 4500  df-bdc 13028  df-bj-ind 13114
This theorem is referenced by:  bj-nnen2lp  13141
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