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Theorem bj-nnelirr 10052
Description: A natural number does not belong to itself. Version of elirr 4266 for natural numbers, which does not require ax-setind 4262. (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 3228 . 2 ¬ ∅ ∈ ∅
2 df-suc 4108 . . . . . 6 suc 𝑦 = (𝑦 ∪ {𝑦})
32eleq2i 2104 . . . . 5 (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4 elun 3084 . . . . . 6 (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5 bj-nntrans 10050 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦 → suc 𝑦𝑦))
6 sucssel 4161 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
75, 6syld 40 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
8 vex 2560 . . . . . . . . . 10 𝑦 ∈ V
98sucid 4154 . . . . . . . . 9 𝑦 ∈ suc 𝑦
10 elsni 3393 . . . . . . . . 9 (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
119, 10syl5eleq 2126 . . . . . . . 8 (suc 𝑦 ∈ {𝑦} → 𝑦𝑦)
1211a1i 9 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦𝑦))
137, 12jaod 637 . . . . . 6 (𝑦 ∈ ω → ((suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}) → 𝑦𝑦))
144, 13syl5bi 141 . . . . 5 (𝑦 ∈ ω → (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) → 𝑦𝑦))
153, 14syl5bi 141 . . . 4 (𝑦 ∈ ω → (suc 𝑦 ∈ suc 𝑦𝑦𝑦))
1615con3d 561 . . 3 (𝑦 ∈ ω → (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
1716rgen 2374 . 2 𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18 ax-bdel 9915 . . . 4 BOUNDED 𝑥𝑥
1918ax-bdn 9911 . . 3 BOUNDED ¬ 𝑥𝑥
20 nfv 1421 . . 3 𝑥 ¬ ∅ ∈ ∅
21 nfv 1421 . . 3 𝑥 ¬ 𝑦𝑦
22 nfv 1421 . . 3 𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23 eleq1 2100 . . . . . 6 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ 𝑥))
24 eleq2 2101 . . . . . 6 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2523, 24bitrd 177 . . . . 5 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ ∅))
2625notbid 592 . . . 4 (𝑥 = ∅ → (¬ 𝑥𝑥 ↔ ¬ ∅ ∈ ∅))
2726biimprd 147 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥𝑥))
28 elequ1 1600 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
29 elequ2 1601 . . . . . 6 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
3028, 29bitrd 177 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
3130notbid 592 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
3231biimpd 132 . . 3 (𝑥 = 𝑦 → (¬ 𝑥𝑥 → ¬ 𝑦𝑦))
33 eleq1 2100 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦𝑥))
34 eleq2 2101 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3533, 34bitrd 177 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3635notbid 592 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
3736biimprd 147 . . 3 (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥𝑥))
38 nfcv 2178 . . 3 𝑥𝐴
39 nfv 1421 . . 3 𝑥 ¬ 𝐴𝐴
40 eleq1 2100 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
41 eleq2 2101 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
4240, 41bitrd 177 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
4342notbid 592 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4443biimpd 132 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 → ¬ 𝐴𝐴))
4519, 20, 21, 22, 27, 32, 37, 38, 39, 44bj-bdfindisg 10047 . 2 ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴𝐴))
461, 17, 45mp2an 402 1 (𝐴 ∈ ω → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 629   = wceq 1243  wcel 1393  wral 2306  cun 2915  wss 2917  c0 3224  {csn 3375  suc csuc 4102  ωcom 4313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9907  ax-bdor 9910  ax-bdn 9911  ax-bdal 9912  ax-bdex 9913  ax-bdeq 9914  ax-bdel 9915  ax-bdsb 9916  ax-bdsep 9978  ax-infvn 10040
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9935  df-bj-ind 10025
This theorem is referenced by:  bj-nnen2lp  10053
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