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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 8772). (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 5087 | . 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅)) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
4 | 3, 3 | breq12d 5087 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦)) |
5 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
6 | 5, 5 | breq12d 5087 | . 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦)) |
7 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
8 | 7, 7 | breq12d 5087 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) |
9 | eqid 2738 | . . 3 ⊢ ∅ = ∅ | |
10 | en0 8803 | . . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅) | |
11 | 9, 10 | mpbir 230 | . 2 ⊢ ∅ ≈ ∅ |
12 | en2sn 8831 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦}) | |
13 | 12 | el2v 3440 | . . . . . 6 ⊢ {𝑦} ≈ {𝑦} |
14 | 13 | jctr 525 | . . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦})) |
15 | nnord 7720 | . . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
16 | orddisj 6304 | . . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅) |
18 | 17, 17 | jca 512 | . . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) |
19 | unen 8836 | . . . . 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 6272 | . . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦}) | |
22 | 20, 21, 21 | 3brtr4g 5108 | . . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦) |
23 | 22 | ex 413 | . 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦)) |
24 | 2, 4, 6, 8, 11, 23 | finds 7745 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 {csn 4561 class class class wbr 5074 Ord word 6265 suc csuc 6268 ωcom 7712 ≈ cen 8730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-om 7713 df-en 8734 |
This theorem is referenced by: nnfi 8950 pssnn 8951 ssnnfiOLD 8953 phplem1 8990 nneneq 8992 onomeneq 9011 onfin 9013 |
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