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Theorem bj-nnelirr 10444
Description: A natural number does not belong to itself. Version of elirr 4293 for natural numbers, which does not require ax-setind 4289. (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 3255 . 2 ¬ ∅ ∈ ∅
2 df-suc 4135 . . . . . 6 suc 𝑦 = (𝑦 ∪ {𝑦})
32eleq2i 2120 . . . . 5 (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4 elun 3111 . . . . . 6 (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5 bj-nntrans 10442 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦 → suc 𝑦𝑦))
6 sucssel 4188 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
75, 6syld 44 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
8 vex 2577 . . . . . . . . . 10 𝑦 ∈ V
98sucid 4181 . . . . . . . . 9 𝑦 ∈ suc 𝑦
10 elsni 3420 . . . . . . . . 9 (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
119, 10syl5eleq 2142 . . . . . . . 8 (suc 𝑦 ∈ {𝑦} → 𝑦𝑦)
1211a1i 9 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦𝑦))
137, 12jaod 647 . . . . . 6 (𝑦 ∈ ω → ((suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}) → 𝑦𝑦))
144, 13syl5bi 145 . . . . 5 (𝑦 ∈ ω → (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) → 𝑦𝑦))
153, 14syl5bi 145 . . . 4 (𝑦 ∈ ω → (suc 𝑦 ∈ suc 𝑦𝑦𝑦))
1615con3d 571 . . 3 (𝑦 ∈ ω → (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
1716rgen 2391 . 2 𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18 ax-bdel 10307 . . . 4 BOUNDED 𝑥𝑥
1918ax-bdn 10303 . . 3 BOUNDED ¬ 𝑥𝑥
20 nfv 1437 . . 3 𝑥 ¬ ∅ ∈ ∅
21 nfv 1437 . . 3 𝑥 ¬ 𝑦𝑦
22 nfv 1437 . . 3 𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23 eleq1 2116 . . . . . 6 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ 𝑥))
24 eleq2 2117 . . . . . 6 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2523, 24bitrd 181 . . . . 5 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ ∅))
2625notbid 602 . . . 4 (𝑥 = ∅ → (¬ 𝑥𝑥 ↔ ¬ ∅ ∈ ∅))
2726biimprd 151 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥𝑥))
28 elequ1 1616 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
29 elequ2 1617 . . . . . 6 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
3028, 29bitrd 181 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
3130notbid 602 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
3231biimpd 136 . . 3 (𝑥 = 𝑦 → (¬ 𝑥𝑥 → ¬ 𝑦𝑦))
33 eleq1 2116 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦𝑥))
34 eleq2 2117 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3533, 34bitrd 181 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3635notbid 602 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
3736biimprd 151 . . 3 (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥𝑥))
38 nfcv 2194 . . 3 𝑥𝐴
39 nfv 1437 . . 3 𝑥 ¬ 𝐴𝐴
40 eleq1 2116 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
41 eleq2 2117 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
4240, 41bitrd 181 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
4342notbid 602 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4443biimpd 136 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 → ¬ 𝐴𝐴))
4519, 20, 21, 22, 27, 32, 37, 38, 39, 44bj-bdfindisg 10439 . 2 ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴𝐴))
461, 17, 45mp2an 410 1 (𝐴 ∈ ω → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 639   = wceq 1259  wcel 1409  wral 2323  cun 2942  wss 2944  c0 3251  {csn 3402  suc csuc 4129  ωcom 4340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910  ax-pr 3971  ax-un 4197  ax-bd0 10299  ax-bdor 10302  ax-bdn 10303  ax-bdal 10304  ax-bdex 10305  ax-bdeq 10306  ax-bdel 10307  ax-bdsb 10308  ax-bdsep 10370  ax-infvn 10432
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-suc 4135  df-iom 4341  df-bdc 10327  df-bj-ind 10417
This theorem is referenced by:  bj-nnen2lp  10445
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