|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > enrefnn | Structured version Visualization version GIF version | ||
| Description: Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 9025). (Contributed by BTernaryTau, 31-Jul-2024.) | 
| Ref | Expression | 
|---|---|
| enrefnn | ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . 3 ⊢ (𝑥 = ∅ → 𝑥 = ∅) | |
| 2 | 1, 1 | breq12d 5155 | . 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅)) | 
| 3 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 4 | 3, 3 | breq12d 5155 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦)) | 
| 5 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
| 6 | 5, 5 | breq12d 5155 | . 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦)) | 
| 7 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 8 | 7, 7 | breq12d 5155 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) | 
| 9 | eqid 2736 | . . 3 ⊢ ∅ = ∅ | |
| 10 | en0 9059 | . . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅) | |
| 11 | 9, 10 | mpbir 231 | . 2 ⊢ ∅ ≈ ∅ | 
| 12 | en2sn 9082 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦}) | |
| 13 | 12 | el2v 3486 | . . . . . 6 ⊢ {𝑦} ≈ {𝑦} | 
| 14 | 13 | jctr 524 | . . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦})) | 
| 15 | nnord 7896 | . . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
| 16 | orddisj 6421 | . . . . . . 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 9087 | . . . . 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 6389 | . . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦}) | |
| 22 | 20, 21, 21 | 3brtr4g 5176 | . . 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 7919 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 {csn 4625 class class class wbr 5142 Ord word 6382 suc csuc 6385 ωcom 7888 ≈ cen 8983 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-om 7889 df-en 8987 | 
| This theorem is referenced by: nnfi 9208 pssnn 9209 phplem1 9245 nneneq 9247 onomeneq 9266 onfin 9268 isinf 9297 | 
| Copyright terms: Public domain | W3C validator |