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Theorem enrefnn 9086
Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 9023). (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 5161 . 2 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ≈ ∅))
3 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43, 3breq12d 5161 . 2 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
5 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
65, 5breq12d 5161 . 2 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87, 7breq12d 5161 . 2 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
9 eqid 2735 . . 3 ∅ = ∅
10 en0 9057 . . 3 (∅ ≈ ∅ ↔ ∅ = ∅)
119, 10mpbir 231 . 2 ∅ ≈ ∅
12 en2sn 9080 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
1312el2v 3485 . . . . . 6 {𝑦} ≈ {𝑦}
1413jctr 524 . . . . 5 (𝑦𝑦 → (𝑦𝑦 ∧ {𝑦} ≈ {𝑦}))
15 nnord 7895 . . . . . . 7 (𝑦 ∈ ω → Ord 𝑦)
16 orddisj 6424 . . . . . . 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 9085 . . . . 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 6392 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
2220, 21, 213brtr4g 5182 . . 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 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  c0 4339  {csn 4631   class class class wbr 5148  Ord word 6385  suc csuc 6388  ωcom 7887  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-om 7888  df-en 8985
This theorem is referenced by:  nnfi  9206  pssnn  9207  phplem1  9242  nneneq  9244  onomeneq  9263  onfin  9265  isinf  9294
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