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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 9737. (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 8782 | . . 3 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅) | |
2 | n0 4312 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (ω ≼ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴) |
4 | reldom 8517 | . . . . . 6 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5611 | . . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | difexg 5233 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ∈ V) |
8 | 7 | adantr 483 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ∈ V) |
9 | simpr 487 | . . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴) | |
10 | difsnpss 4742 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴) | |
11 | 9, 10 | sylib 220 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ⊊ 𝐴) |
12 | infdifsn 9122 | . . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ≈ 𝐴) | |
13 | 12 | adantr 483 | . . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ≈ 𝐴) |
14 | 11, 13 | jca 514 | . . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴)) |
15 | psseq1 4066 | . . . 4 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ⊊ 𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴)) | |
16 | breq1 5071 | . . . 4 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥 ≈ 𝐴 ↔ (𝐴 ∖ {𝑦}) ≈ 𝐴)) | |
17 | 15, 16 | anbi12d 632 | . . 3 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴) ↔ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴))) |
18 | 8, 14, 17 | spcedv 3601 | . 2 ⊢ ((ω ≼ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
19 | 3, 18 | exlimddv 1936 | 1 ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∖ cdif 3935 ⊊ wpss 3939 ∅c0 4293 {csn 4569 class class class wbr 5068 ωcom 7582 ≈ cen 8508 ≼ cdom 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 |
This theorem is referenced by: isfin4-2 9738 |
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