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Theorem enrefnn 8647
Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8589). (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 5043 . 2 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ≈ ∅))
3 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43, 3breq12d 5043 . 2 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
5 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
65, 5breq12d 5043 . 2 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87, 7breq12d 5043 . 2 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
9 eqid 2738 . . 3 ∅ = ∅
10 en0 8620 . . 3 (∅ ≈ ∅ ↔ ∅ = ∅)
119, 10mpbir 234 . 2 ∅ ≈ ∅
12 en2sn 8642 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
1312el2v 3406 . . . . . 6 {𝑦} ≈ {𝑦}
1413jctr 528 . . . . 5 (𝑦𝑦 → (𝑦𝑦 ∧ {𝑦} ≈ {𝑦}))
15 nnord 7609 . . . . . . 7 (𝑦 ∈ ω → Ord 𝑦)
16 orddisj 6210 . . . . . . 7 (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
1715, 16syl 17 . . . . . 6 (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
1817, 17jca 515 . . . . 5 (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19 unen 8646 . . . . 5 (((𝑦𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
2014, 18, 19syl2anr 600 . . . 4 ((𝑦 ∈ ω ∧ 𝑦𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21 df-suc 6178 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
2220, 21, 213brtr4g 5064 . . 3 ((𝑦 ∈ ω ∧ 𝑦𝑦) → suc 𝑦 ≈ suc 𝑦)
2322ex 416 . 2 (𝑦 ∈ ω → (𝑦𝑦 → suc 𝑦 ≈ suc 𝑦))
242, 4, 6, 8, 11, 23finds 7631 1 (𝐴 ∈ ω → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  Vcvv 3398  cun 3841  cin 3842  c0 4211  {csn 4516   class class class wbr 5030  Ord word 6171  suc csuc 6174  ωcom 7601  cen 8554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-om 7602  df-en 8558
This theorem is referenced by:  nnfi  8768  pssnn  8769  ssnnfiOLD  8771
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