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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 8589). (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 5043 | . 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅)) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
4 | 3, 3 | breq12d 5043 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦)) |
5 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
6 | 5, 5 | breq12d 5043 | . 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦)) |
7 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
8 | 7, 7 | breq12d 5043 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) |
9 | eqid 2738 | . . 3 ⊢ ∅ = ∅ | |
10 | en0 8620 | . . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅) | |
11 | 9, 10 | mpbir 234 | . 2 ⊢ ∅ ≈ ∅ |
12 | en2sn 8642 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦}) | |
13 | 12 | el2v 3406 | . . . . . 6 ⊢ {𝑦} ≈ {𝑦} |
14 | 13 | jctr 528 | . . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦})) |
15 | nnord 7609 | . . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
16 | orddisj 6210 | . . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅) |
18 | 17, 17 | jca 515 | . . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) |
19 | unen 8646 | . . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦})) | |
20 | 14, 18, 19 | syl2anr 600 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦})) |
21 | df-suc 6178 | . . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦}) | |
22 | 20, 21, 21 | 3brtr4g 5064 | . . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦) |
23 | 22 | ex 416 | . 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦)) |
24 | 2, 4, 6, 8, 11, 23 | finds 7631 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∪ cun 3841 ∩ cin 3842 ∅c0 4211 {csn 4516 class class class wbr 5030 Ord word 6171 suc csuc 6174 ωcom 7601 ≈ cen 8554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-om 7602 df-en 8558 |
This theorem is referenced by: nnfi 8768 pssnn 8769 ssnnfiOLD 8771 |
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