Theorem enrefnn 8702

Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8638). (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 5052	. 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅))
3		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	3, 3	breq12d 5052	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
6	5, 5	breq12d 5052	. 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
8	7, 7	breq12d 5052	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴))
9		eqid 2736	. . 3 ⊢ ∅ = ∅
10		en0 8669	. . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅)
11	9, 10	mpbir 234	. 2 ⊢ ∅ ≈ ∅
12		en2sn 8696	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
13	12	el2v 3406	. . . . . 6 ⊢ {𝑦} ≈ {𝑦}
14	13	jctr 528	. . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}))
15		nnord 7630	. . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦)
16		orddisj 6229	. . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
18	17, 17	jca 515	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19		unen 8701	. . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
20	14, 18, 19	syl2anr 600	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21		df-suc 6197	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
22	20, 21, 21	3brtr4g 5073	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦)
23	22	ex 416	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦))
24	2, 4, 6, 8, 11, 23	finds 7654	1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ∪ cun 3851   ∩ cin 3852  ∅c0 4223  {csn 4527   class class class wbr 5039  Ord word 6190  suc csuc 6193  ωcom 7622   ≈ cen 8601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-om 7623  df-en 8605

This theorem is referenced by:  nnfi  8823  pssnn  8824  ssnnfiOLD  8826