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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 8982). (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 5160 | . 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅)) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
4 | 3, 3 | breq12d 5160 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦)) |
5 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
6 | 5, 5 | breq12d 5160 | . 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦)) |
7 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
8 | 7, 7 | breq12d 5160 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) |
9 | eqid 2730 | . . 3 ⊢ ∅ = ∅ | |
10 | en0 9015 | . . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅) | |
11 | 9, 10 | mpbir 230 | . 2 ⊢ ∅ ≈ ∅ |
12 | en2sn 9043 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦}) | |
13 | 12 | el2v 3480 | . . . . . 6 ⊢ {𝑦} ≈ {𝑦} |
14 | 13 | jctr 523 | . . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦})) |
15 | nnord 7865 | . . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
16 | orddisj 6401 | . . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅) |
18 | 17, 17 | jca 510 | . . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) |
19 | unen 9048 | . . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦})) | |
20 | 14, 18, 19 | syl2anr 595 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦})) |
21 | df-suc 6369 | . . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦}) | |
22 | 20, 21, 21 | 3brtr4g 5181 | . . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦) |
23 | 22 | ex 411 | . 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦)) |
24 | 2, 4, 6, 8, 11, 23 | finds 7891 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∪ cun 3945 ∩ cin 3946 ∅c0 4321 {csn 4627 class class class wbr 5147 Ord word 6362 suc csuc 6365 ωcom 7857 ≈ cen 8938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-om 7858 df-en 8942 |
This theorem is referenced by: nnfi 9169 pssnn 9170 ssnnfiOLD 9172 phplem1 9209 nneneq 9211 onomeneq 9230 onfin 9232 isinf 9262 |
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