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Mirrors > Home > MPE Home > Th. List > pssinf | Structured version Visualization version GIF version |
Description: A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) |
Ref | Expression |
---|---|
pssinf | ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐴 ≈ 𝐵) → ¬ 𝐵 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | php3 9237 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊊ 𝐵) → 𝐴 ≺ 𝐵) | |
2 | 1 | ex 412 | . . . 4 ⊢ (𝐵 ∈ Fin → (𝐴 ⊊ 𝐵 → 𝐴 ≺ 𝐵)) |
3 | sdomnen 9002 | . . . 4 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | |
4 | 2, 3 | syl6com 37 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∈ Fin → ¬ 𝐴 ≈ 𝐵)) |
5 | 4 | con2d 134 | . 2 ⊢ (𝐴 ⊊ 𝐵 → (𝐴 ≈ 𝐵 → ¬ 𝐵 ∈ Fin)) |
6 | 5 | imp 406 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐴 ≈ 𝐵) → ¬ 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2099 ⊊ wpss 3948 class class class wbr 5148 ≈ cen 8961 ≺ csdm 8963 Fincfn 8964 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-1o 8487 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 |
This theorem is referenced by: fisseneq 9282 ominf 9283 ominfOLD 9284 |
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