Theorem enrefnn 9047

Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8980). (Contributed by BTernaryTau, 31-Jul-2024.)

Assertion

Ref	Expression
enrefnn	⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴)

Proof of Theorem enrefnn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
2	1, 1	breq12d 5162	. 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅))
3		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	3, 3	breq12d 5162	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
6	5, 5	breq12d 5162	. 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
8	7, 7	breq12d 5162	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴))
9		eqid 2733	. . 3 ⊢ ∅ = ∅
10		en0 9013	. . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅)
11	9, 10	mpbir 230	. 2 ⊢ ∅ ≈ ∅
12		en2sn 9041	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
13	12	el2v 3483	. . . . . 6 ⊢ {𝑦} ≈ {𝑦}
14	13	jctr 526	. . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}))
15		nnord 7863	. . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦)
16		orddisj 6403	. . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
18	17, 17	jca 513	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19		unen 9046	. . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
20	14, 18, 19	syl2anr 598	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21		df-suc 6371	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
22	20, 21, 21	3brtr4g 5183	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦)
23	22	ex 414	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦))
24	2, 4, 6, 8, 11, 23	finds 7889	1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∪ cun 3947   ∩ cin 3948  ∅c0 4323  {csn 4629   class class class wbr 5149  Ord word 6364  suc csuc 6367  ωcom 7855   ≈ cen 8936

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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

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 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-om 7856  df-en 8940

This theorem is referenced by:  nnfi  9167  pssnn  9168  ssnnfiOLD  9170  phplem1  9207  nneneq  9209  onomeneq  9228  onfin  9230  isinf  9260