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| Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 10353. (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 9340 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) | |
| 2 | n0 4353 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) | 
| 4 | reldom 8991 | . . . . . 6 ⊢ Rel ≼ | |
| 5 | 4 | brrelex2i 5742 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) | 
| 6 | 5 | difexd 5331 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ∈ V) | 
| 7 | 6 | adantr 480 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ∈ V) | 
| 8 | simpr 484 | . . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
| 9 | difsnpss 4807 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴) | |
| 10 | 8, 9 | sylib 218 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ⊊ 𝐴) | 
| 11 | infdifsn 9697 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ≈ 𝐴) | |
| 12 | 11 | adantr 480 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ≈ 𝐴) | 
| 13 | 10, 12 | jca 511 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴)) | 
| 14 | psseq1 4090 | . . . 4 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ⊊ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴)) | |
| 15 | breq1 5146 | . . . 4 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ≈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ≈ 𝐴)) | |
| 16 | 14, 15 | anbi12d 632 | . . 3 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴))) | 
| 17 | 7, 13, 16 | spcedv 3598 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) | 
| 18 | 3, 17 | exlimddv 1935 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ⊊ wpss 3952 ∅c0 4333 {csn 4626 class class class wbr 5143 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-er 8745 df-en 8986 df-dom 8987 | 
| This theorem is referenced by: isfin4-2 10354 | 
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