Theorem enrefnn 9053

Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8990). (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 23	. . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
2	1, 1	breq12d 5127	. 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅))
3		id 23	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	3, 3	breq12d 5127	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦))
5		id 23	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
6	5, 5	breq12d 5127	. 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7		id 23	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
8	7, 7	breq12d 5127	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴))
9		eqid 2766	. . 3 ⊢ ∅ = ∅
10		en0 9024	. . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅)
11	9, 10	mpbir 234	. 2 ⊢ ∅ ≈ ∅
12		en2sn 9048	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
13	12	el2v 3465	. . . . . 6 ⊢ {𝑦} ≈ {𝑦}
14	13	jctr 534	. . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}))
15		nnord 7879	. . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦)
16		orddisj 6406	. . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
17	15, 16	syl 18	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
18	17, 17	jca 521	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19		unen 9052	. . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
20	14, 18, 19	syl2anr 609	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21		df-suc 6373	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
22	20, 21, 21	3brtr4g 5150	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦)
23	22	ex 418	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦))
24	2, 4, 6, 8, 11, 23	finds 7902	1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∪ cun 3906   ∩ cin 3907  ∅c0 4289  {csn 4594   class class class wbr 5114  Ord word 6366  suc csuc 6369  ωcom 7871   ≈ cen 8949

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-om 7872  df-en 8953

This theorem is used by:  nnfi  9162  pssnn  9163  phplem1  9198  nneneq  9200  onomeneq  9208  onfin  9209  isinf  9235  kardnnfi  35606