Theorem enrefnn 8983

Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8921). (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 5111	. 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅))
3		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	3, 3	breq12d 5111	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
6	5, 5	breq12d 5111	. 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
8	7, 7	breq12d 5111	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴))
9		eqid 2736	. . 3 ⊢ ∅ = ∅
10		en0 8955	. . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅)
11	9, 10	mpbir 231	. 2 ⊢ ∅ ≈ ∅
12		en2sn 8978	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
13	12	el2v 3447	. . . . . 6 ⊢ {𝑦} ≈ {𝑦}
14	13	jctr 524	. . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}))
15		nnord 7816	. . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦)
16		orddisj 6355	. . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅)
18	17, 17	jca 511	. . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅))
19		unen 8982	. . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
20	14, 18, 19	syl2anr 597	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21		df-suc 6323	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
22	20, 21, 21	3brtr4g 5132	. . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦)
23	22	ex 412	. 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦))
24	2, 4, 6, 8, 11, 23	finds 7838	1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899   ∩ cin 3900  ∅c0 4285  {csn 4580   class class class wbr 5098  Ord word 6316  suc csuc 6319  ωcom 7808   ≈ cen 8880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-om 7809  df-en 8884

This theorem is referenced by:  nnfi  9092  pssnn  9093  phplem1  9128  nneneq  9130  onomeneq  9138  onfin  9139  isinf  9165