Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > infpss | Structured version Visualization version GIF version |
Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9135. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
infpss | ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infn0 8222 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) | |
2 | n0 3931 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylib 208 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) |
4 | reldom 7961 | . . . . . 6 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5159 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | difexg 4808 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ∈ V) |
8 | 7 | adantr 481 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ∈ V) |
9 | simpr 477 | . . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
10 | difsnpss 4338 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴) | |
11 | 9, 10 | sylib 208 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ⊊ 𝐴) |
12 | infdifsn 8554 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ≈ 𝐴) | |
13 | 12 | adantr 481 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ≈ 𝐴) |
14 | 11, 13 | jca 554 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴)) |
15 | psseq1 3694 | . . . . 5 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ⊊ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴)) | |
16 | breq1 4656 | . . . . 5 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ≈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ≈ 𝐴)) | |
17 | 15, 16 | anbi12d 747 | . . . 4 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴))) |
18 | 17 | spcegv 3294 | . . 3 ⊢ ((𝐴 ∖ {𝑦}) ∈ V → (((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴) → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴))) |
19 | 8, 14, 18 | sylc 65 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
20 | 3, 19 | exlimddv 1863 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ⊊ wpss 3575 ∅c0 3915 {csn 4177 class class class wbr 4653 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 |
This theorem is referenced by: isfin4-2 9136 |
Copyright terms: Public domain | W3C validator |