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Theorem bj-nnelirr 13728
Description: A natural number does not belong to itself. Version of elirr 4515 for natural numbers, which does not require ax-setind 4511. (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 3411 . 2 ¬ ∅ ∈ ∅
2 df-suc 4346 . . . . . 6 suc 𝑦 = (𝑦 ∪ {𝑦})
32eleq2i 2231 . . . . 5 (suc 𝑦 ∈ suc 𝑦 ↔ suc 𝑦 ∈ (𝑦 ∪ {𝑦}))
4 elun 3261 . . . . . 6 (suc 𝑦 ∈ (𝑦 ∪ {𝑦}) ↔ (suc 𝑦𝑦 ∨ suc 𝑦 ∈ {𝑦}))
5 bj-nntrans 13726 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦 → suc 𝑦𝑦))
6 sucssel 4399 . . . . . . . 8 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
75, 6syld 45 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦𝑦𝑦𝑦))
8 vex 2727 . . . . . . . . . 10 𝑦 ∈ V
98sucid 4392 . . . . . . . . 9 𝑦 ∈ suc 𝑦
10 elsni 3591 . . . . . . . . 9 (suc 𝑦 ∈ {𝑦} → suc 𝑦 = 𝑦)
119, 10eleqtrid 2253 . . . . . . . 8 (suc 𝑦 ∈ {𝑦} → 𝑦𝑦)
1211a1i 9 . . . . . . 7 (𝑦 ∈ ω → (suc 𝑦 ∈ {𝑦} → 𝑦𝑦))
137, 12jaod 707 . . . . . 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 621 . . 3 (𝑦 ∈ ω → (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦))
1716rgen 2517 . 2 𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
18 ax-bdel 13596 . . . 4 BOUNDED 𝑥𝑥
1918ax-bdn 13592 . . 3 BOUNDED ¬ 𝑥𝑥
20 nfv 1515 . . 3 𝑥 ¬ ∅ ∈ ∅
21 nfv 1515 . . 3 𝑥 ¬ 𝑦𝑦
22 nfv 1515 . . 3 𝑥 ¬ suc 𝑦 ∈ suc 𝑦
23 eleq1 2227 . . . . . 6 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ 𝑥))
24 eleq2 2228 . . . . . 6 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2523, 24bitrd 187 . . . . 5 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ∈ ∅))
2625notbid 657 . . . 4 (𝑥 = ∅ → (¬ 𝑥𝑥 ↔ ¬ ∅ ∈ ∅))
2726biimprd 157 . . 3 (𝑥 = ∅ → (¬ ∅ ∈ ∅ → ¬ 𝑥𝑥))
28 elequ1 2139 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
29 elequ2 2140 . . . . . 6 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
3028, 29bitrd 187 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
3130notbid 657 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
3231biimpd 143 . . 3 (𝑥 = 𝑦 → (¬ 𝑥𝑥 → ¬ 𝑦𝑦))
33 eleq1 2227 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦𝑥))
34 eleq2 2228 . . . . . 6 (𝑥 = suc 𝑦 → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3533, 34bitrd 187 . . . . 5 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ∈ suc 𝑦))
3635notbid 657 . . . 4 (𝑥 = suc 𝑦 → (¬ 𝑥𝑥 ↔ ¬ suc 𝑦 ∈ suc 𝑦))
3736biimprd 157 . . 3 (𝑥 = suc 𝑦 → (¬ suc 𝑦 ∈ suc 𝑦 → ¬ 𝑥𝑥))
38 nfcv 2306 . . 3 𝑥𝐴
39 nfv 1515 . . 3 𝑥 ¬ 𝐴𝐴
40 eleq1 2227 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝑥))
41 eleq2 2228 . . . . . 6 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
4240, 41bitrd 187 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
4342notbid 657 . . . 4 (𝑥 = 𝐴 → (¬ 𝑥𝑥 ↔ ¬ 𝐴𝐴))
4443biimpd 143 . . 3 (𝑥 = 𝐴 → (¬ 𝑥𝑥 → ¬ 𝐴𝐴))
4519, 20, 21, 22, 27, 32, 37, 38, 39, 44bj-bdfindisg 13723 . 2 ((¬ ∅ ∈ ∅ ∧ ∀𝑦 ∈ ω (¬ 𝑦𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)) → (𝐴 ∈ ω → ¬ 𝐴𝐴))
461, 17, 45mp2an 423 1 (𝐴 ∈ ω → ¬ 𝐴𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 698   = wceq 1342  wcel 2135  wral 2442  cun 3112  wss 3114  c0 3407  {csn 3573  suc csuc 4340  ωcom 4564
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-nul 4105  ax-pr 4184  ax-un 4408  ax-bd0 13588  ax-bdor 13591  ax-bdn 13592  ax-bdal 13593  ax-bdex 13594  ax-bdeq 13595  ax-bdel 13596  ax-bdsb 13597  ax-bdsep 13659  ax-infvn 13716
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-sn 3579  df-pr 3580  df-uni 3787  df-int 3822  df-suc 4346  df-iom 4565  df-bdc 13616  df-bj-ind 13702
This theorem is referenced by:  bj-nnen2lp  13729
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