Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > enfiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of enfi 8842 as of 23-Sep-2024. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
enfiOLD | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enen1 8764 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥)) | |
2 | 1 | rexbidv 3206 | . 2 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)) |
3 | isfi 8630 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
4 | isfi 8630 | . 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) | |
5 | 2, 3, 4 | 3bitr4g 317 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 ∃wrex 3052 class class class wbr 5039 ωcom 7622 ≈ cen 8601 Fincfn 8604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-er 8369 df-en 8605 df-fin 8608 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |