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Theorem enrefnn 8837
Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8772). (Contributed by BTernaryTau, 31-Jul-2024.)
Assertion
Ref Expression
enrefnn (𝐴 ∈ ω → 𝐴𝐴)

Proof of Theorem enrefnn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝑥 = ∅ → 𝑥 = ∅)
21, 1breq12d 5087 . 2 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ≈ ∅))
3 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43, 3breq12d 5087 . 2 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
5 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
65, 5breq12d 5087 . 2 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87, 7breq12d 5087 . 2 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
9 eqid 2738 . . 3 ∅ = ∅
10 en0 8803 . . 3 (∅ ≈ ∅ ↔ ∅ = ∅)
119, 10mpbir 230 . 2 ∅ ≈ ∅
12 en2sn 8831 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
1312el2v 3440 . . . . . 6 {𝑦} ≈ {𝑦}
1413jctr 525 . . . . 5 (𝑦𝑦 → (𝑦𝑦 ∧ {𝑦} ≈ {𝑦}))
15 nnord 7720 . . . . . . 7 (𝑦 ∈ ω → Ord 𝑦)
16 orddisj 6304 . . . . . . 7 (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
1715, 16syl 17 . . . . . 6 (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
1817, 17jca 512 . . . . 5 (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19 unen 8836 . . . . 5 (((𝑦𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
2014, 18, 19syl2anr 597 . . . 4 ((𝑦 ∈ ω ∧ 𝑦𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21 df-suc 6272 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
2220, 21, 213brtr4g 5108 . . 3 ((𝑦 ∈ ω ∧ 𝑦𝑦) → suc 𝑦 ≈ suc 𝑦)
2322ex 413 . 2 (𝑦 ∈ ω → (𝑦𝑦 → suc 𝑦 ≈ suc 𝑦))
242, 4, 6, 8, 11, 23finds 7745 1 (𝐴 ∈ ω → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cun 3885  cin 3886  c0 4256  {csn 4561   class class class wbr 5074  Ord word 6265  suc csuc 6268  ωcom 7712  cen 8730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-om 7713  df-en 8734
This theorem is referenced by:  nnfi  8950  pssnn  8951  ssnnfiOLD  8953  phplem1  8990  nneneq  8992  onomeneq  9011  onfin  9013
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