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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 9023). (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 5161 | . 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅)) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
4 | 3, 3 | breq12d 5161 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦)) |
5 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
6 | 5, 5 | breq12d 5161 | . 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦)) |
7 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
8 | 7, 7 | breq12d 5161 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) |
9 | eqid 2735 | . . 3 ⊢ ∅ = ∅ | |
10 | en0 9057 | . . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅) | |
11 | 9, 10 | mpbir 231 | . 2 ⊢ ∅ ≈ ∅ |
12 | en2sn 9080 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦}) | |
13 | 12 | el2v 3485 | . . . . . 6 ⊢ {𝑦} ≈ {𝑦} |
14 | 13 | jctr 524 | . . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦})) |
15 | nnord 7895 | . . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
16 | orddisj 6424 | . . . . . . 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 9085 | . . . . 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 6392 | . . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦}) | |
22 | 20, 21, 21 | 3brtr4g 5182 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 class class class wbr 5148 Ord word 6385 suc csuc 6388 ωcom 7887 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-om 7888 df-en 8985 |
This theorem is referenced by: nnfi 9206 pssnn 9207 phplem1 9242 nneneq 9244 onomeneq 9263 onfin 9265 isinf 9294 |
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