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Theorem enrefnn 8998
Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8931). (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 5123 . 2 (𝑥 = ∅ → (𝑥𝑥 ↔ ∅ ≈ ∅))
3 id 22 . . 3 (𝑥 = 𝑦𝑥 = 𝑦)
43, 3breq12d 5123 . 2 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
5 id 22 . . 3 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
65, 5breq12d 5123 . 2 (𝑥 = suc 𝑦 → (𝑥𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
87, 7breq12d 5123 . 2 (𝑥 = 𝐴 → (𝑥𝑥𝐴𝐴))
9 eqid 2737 . . 3 ∅ = ∅
10 en0 8964 . . 3 (∅ ≈ ∅ ↔ ∅ = ∅)
119, 10mpbir 230 . 2 ∅ ≈ ∅
12 en2sn 8992 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
1312el2v 3456 . . . . . 6 {𝑦} ≈ {𝑦}
1413jctr 526 . . . . 5 (𝑦𝑦 → (𝑦𝑦 ∧ {𝑦} ≈ {𝑦}))
15 nnord 7815 . . . . . . 7 (𝑦 ∈ ω → Ord 𝑦)
16 orddisj 6360 . . . . . . 7 (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
1715, 16syl 17 . . . . . 6 (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
1817, 17jca 513 . . . . 5 (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19 unen 8997 . . . . 5 (((𝑦𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
2014, 18, 19syl2anr 598 . . . 4 ((𝑦 ∈ ω ∧ 𝑦𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21 df-suc 6328 . . . 4 suc 𝑦 = (𝑦 ∪ {𝑦})
2220, 21, 213brtr4g 5144 . . 3 ((𝑦 ∈ ω ∧ 𝑦𝑦) → suc 𝑦 ≈ suc 𝑦)
2322ex 414 . 2 (𝑦 ∈ ω → (𝑦𝑦 → suc 𝑦 ≈ suc 𝑦))
242, 4, 6, 8, 11, 23finds 7840 1 (𝐴 ∈ ω → 𝐴𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3448  cun 3913  cin 3914  c0 4287  {csn 4591   class class class wbr 5110  Ord word 6321  suc csuc 6324  ωcom 7807  cen 8887
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-om 7808  df-en 8891
This theorem is referenced by:  nnfi  9118  pssnn  9119  ssnnfiOLD  9121  phplem1  9158  nneneq  9160  onomeneq  9179  onfin  9181  isinf  9211
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