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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 10079. (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 9063 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) | |
2 | n0 4280 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylib 217 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) |
4 | reldom 8726 | . . . . . 6 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5639 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | 5 | difexd 5251 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ∈ V) |
7 | 6 | adantr 481 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ∈ V) |
8 | simpr 485 | . . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
9 | difsnpss 4740 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴) | |
10 | 8, 9 | sylib 217 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ⊊ 𝐴) |
11 | infdifsn 9402 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ≈ 𝐴) | |
12 | 11 | adantr 481 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ≈ 𝐴) |
13 | 10, 12 | jca 512 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴)) |
14 | psseq1 4021 | . . . 4 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ⊊ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴)) | |
15 | breq1 5076 | . . . 4 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ≈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ≈ 𝐴)) | |
16 | 14, 15 | anbi12d 631 | . . 3 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴))) |
17 | 7, 13, 16 | spcedv 3534 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
18 | 3, 17 | exlimddv 1938 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 Vcvv 3429 ∖ cdif 3883 ⊊ wpss 3887 ∅c0 4256 {csn 4561 class class class wbr 5073 ωcom 7702 ≈ cen 8717 ≼ cdom 8718 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-om 7703 df-1o 8284 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 |
This theorem is referenced by: isfin4-2 10080 |
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