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Theorem bj-nnelirr 13078
Description: A natural number does not belong to itself. Version of elirr 4426 for natural numbers, which does not require ax-setind 4422. (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 3337 . 2 ¬ ∅ ∈ ∅
2 df-suc 4263 . . . . . 6 suc 𝑦 = (𝑦 ∪ {𝑦})
32eleq2i 2184 . . . . 5 (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4 elun 3187 . . . . . 6 (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5 bj-nntrans 13076 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦 → suc 𝑦𝑦))
6 sucssel 4316 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
75, 6syld 45 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
8 vex 2663 . . . . . . . . . 10 𝑦 ∈ V
98sucid 4309 . . . . . . . . 9 𝑦 ∈ suc 𝑦
10 elsni 3515 . . . . . . . . 9 (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
119, 10eleqtrid 2206 . . . . . . . 8 (suc 𝑦 ∈ {𝑦} → 𝑦𝑦)
1211a1i 9 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦𝑦))
137, 12jaod 691 . . . . . 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 605 . . 3 (𝑦 ∈ ω → (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
1716rgen 2462 . 2 𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18 ax-bdel 12946 . . . 4 BOUNDED 𝑥𝑥
1918ax-bdn 12942 . . 3 BOUNDED ¬ 𝑥𝑥
20 nfv 1493 . . 3 𝑥 ¬ ∅ ∈ ∅
21 nfv 1493 . . 3 𝑥 ¬ 𝑦𝑦
22 nfv 1493 . . 3 𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23 eleq1 2180 . . . . . 6 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ 𝑥))
24 eleq2 2181 . . . . . 6 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2523, 24bitrd 187 . . . . 5 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ ∅))
2625notbid 641 . . . 4 (𝑥 = ∅ → (¬ 𝑥𝑥 ↔ ¬ ∅ ∈ ∅))
2726biimprd 157 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥𝑥))
28 elequ1 1675 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
29 elequ2 1676 . . . . . 6 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
3028, 29bitrd 187 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
3130notbid 641 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
3231biimpd 143 . . 3 (𝑥 = 𝑦 → (¬ 𝑥𝑥 → ¬ 𝑦𝑦))
33 eleq1 2180 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦𝑥))
34 eleq2 2181 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3533, 34bitrd 187 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3635notbid 641 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
3736biimprd 157 . . 3 (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥𝑥))
38 nfcv 2258 . . 3 𝑥𝐴
39 nfv 1493 . . 3 𝑥 ¬ 𝐴𝐴
40 eleq1 2180 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
41 eleq2 2181 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
4240, 41bitrd 187 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
4342notbid 641 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4443biimpd 143 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 → ¬ 𝐴𝐴))
4519, 20, 21, 22, 27, 32, 37, 38, 39, 44bj-bdfindisg 13073 . 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 682   = wceq 1316  wcel 1465  wral 2393  cun 3039  wss 3041  c0 3333  {csn 3497  suc csuc 4257  ωcom 4474
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdor 12941  ax-bdn 12942  ax-bdal 12943  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-infvn 13066
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by:  bj-nnen2lp  13079
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