Theorem enrefnn 9113

Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 9044). (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 5179	. 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅))
3		id 22	. . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
4	3, 3	breq12d 5179	. 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦))
5		id 22	. . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
6	5, 5	breq12d 5179	. 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦))
7		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
8	7, 7	breq12d 5179	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴))
9		eqid 2740	. . 3 ⊢ ∅ = ∅
10		en0 9078	. . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅)
11	9, 10	mpbir 231	. 2 ⊢ ∅ ≈ ∅
12		en2sn 9106	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦})
13	12	el2v 3495	. . . . . 6 ⊢ {𝑦} ≈ {𝑦}
14	13	jctr 524	. . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}))
15		nnord 7911	. . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦)
16		orddisj 6433	. . . . . . 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 9112	. . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
20	14, 18, 19	syl2anr 596	. . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦}))
21		df-suc 6401	. . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
22	20, 21, 21	3brtr4g 5200	. . 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 7936	1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  {csn 4648   class class class wbr 5166  Ord word 6394  suc csuc 6397  ωcom 7903   ≈ cen 9000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-om 7904  df-en 9004

This theorem is referenced by:  nnfi  9233  pssnn  9234  ssnnfiOLD  9236  phplem1  9270  nneneq  9272  onomeneq  9291  onfin  9293  isinf  9323