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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 8968). (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 5157 | . 2 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑥 ↔ ∅ ≈ ∅)) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
4 | 3, 3 | breq12d 5157 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑥 ↔ 𝑦 ≈ 𝑦)) |
5 | id 22 | . . 3 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦) | |
6 | 5, 5 | breq12d 5157 | . 2 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑥 ↔ suc 𝑦 ≈ suc 𝑦)) |
7 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
8 | 7, 7 | breq12d 5157 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑥 ↔ 𝐴 ≈ 𝐴)) |
9 | eqid 2733 | . . 3 ⊢ ∅ = ∅ | |
10 | en0 9001 | . . 3 ⊢ (∅ ≈ ∅ ↔ ∅ = ∅) | |
11 | 9, 10 | mpbir 230 | . 2 ⊢ ∅ ≈ ∅ |
12 | en2sn 9029 | . . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑦 ∈ V) → {𝑦} ≈ {𝑦}) | |
13 | 12 | el2v 3483 | . . . . . 6 ⊢ {𝑦} ≈ {𝑦} |
14 | 13 | jctr 526 | . . . . 5 ⊢ (𝑦 ≈ 𝑦 → (𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦})) |
15 | nnord 7850 | . . . . . . 7 ⊢ (𝑦 ∈ ω → Ord 𝑦) | |
16 | orddisj 6394 | . . . . . . 7 ⊢ (Ord 𝑦 → (𝑦 ∩ {𝑦}) = ∅) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝑦 ∈ ω → (𝑦 ∩ {𝑦}) = ∅) |
18 | 17, 17 | jca 513 | . . . . 5 ⊢ (𝑦 ∈ ω → ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) |
19 | unen 9034 | . . . . 5 ⊢ (((𝑦 ≈ 𝑦 ∧ {𝑦} ≈ {𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (𝑦 ∩ {𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦})) | |
20 | 14, 18, 19 | syl2anr 598 | . . . 4 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → (𝑦 ∪ {𝑦}) ≈ (𝑦 ∪ {𝑦})) |
21 | df-suc 6362 | . . . 4 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦}) | |
22 | 20, 21, 21 | 3brtr4g 5178 | . . 3 ⊢ ((𝑦 ∈ ω ∧ 𝑦 ≈ 𝑦) → suc 𝑦 ≈ suc 𝑦) |
23 | 22 | ex 414 | . 2 ⊢ (𝑦 ∈ ω → (𝑦 ≈ 𝑦 → suc 𝑦 ≈ suc 𝑦)) |
24 | 2, 4, 6, 8, 11, 23 | finds 7876 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3944 ∩ cin 3945 ∅c0 4320 {csn 4624 class class class wbr 5144 Ord word 6355 suc csuc 6358 ωcom 7842 ≈ cen 8924 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-om 7843 df-en 8928 |
This theorem is referenced by: nnfi 9155 pssnn 9156 ssnnfiOLD 9158 phplem1 9195 nneneq 9197 onomeneq 9216 onfin 9218 isinf 9248 |
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