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Theorem enrefnn 9088
Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 9025). (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 5155 . 2 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ≈ ∅))
3 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43, 3breq12d 5155 . 2 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
5 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
65, 5breq12d 5155 . 2 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87, 7breq12d 5155 . 2 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
9 eqid 2736 . . 3 ∅ = ∅
10 en0 9059 . . 3 (∅ ≈ ∅ ↔ ∅ = ∅)
119, 10mpbir 231 . 2 ∅ ≈ ∅
12 en2sn 9082 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
1312el2v 3486 . . . . . 6 {𝑦} ≈ {𝑦}
1413jctr 524 . . . . 5 (𝑦𝑦 → (𝑦𝑦 ∧ {𝑦} ≈ {𝑦}))
15 nnord 7896 . . . . . . 7 (𝑦 ∈ ω → Ord 𝑦)
16 orddisj 6421 . . . . . . 7 (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
1715, 16syl 17 . . . . . 6 (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
1817, 17jca 511 . . . . 5 (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19 unen 9087 . . . . 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 6389 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
2220, 21, 213brtr4g 5176 . . 3 ((𝑦 ∈ ω ∧ 𝑦𝑦) → suc 𝑦 ≈ suc 𝑦)
2322ex 412 . 2 (𝑦 ∈ ω → (𝑦𝑦 → suc 𝑦 ≈ suc 𝑦))
242, 4, 6, 8, 11, 23finds 7919 1 (𝐴 ∈ ω → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cun 3948  cin 3949  c0 4332  {csn 4625   class class class wbr 5142  Ord word 6382  suc csuc 6385  ωcom 7888  cen 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-om 7889  df-en 8987
This theorem is referenced by:  nnfi  9208  pssnn  9209  phplem1  9245  nneneq  9247  onomeneq  9266  onfin  9268  isinf  9297
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