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Mirrors > Home > MPE Home > Th. List > enreffi | Structured version Visualization version GIF version |
Description: Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg 8998). (Contributed by BTernaryTau, 8-Sep-2024.) |
Ref | Expression |
---|---|
enreffi | ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6871 | . 2 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
2 | f1oenfi 9200 | . 2 ⊢ ((𝐴 ∈ Fin ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝐴 ≈ 𝐴) | |
3 | 1, 2 | mpan2 690 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5142 I cid 5569 ↾ cres 5674 –1-1-onto→wf1o 6541 ≈ cen 8954 Fincfn 8957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 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-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7865 df-1o 8480 df-en 8958 df-fin 8961 |
This theorem is referenced by: fidomndrnglem 21253 |
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