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Mirrors > Home > ILE Home > Th. List > phpelm | GIF version |
Description: Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.) |
Ref | Expression |
---|---|
phpelm | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ ω) | |
2 | nnon 4350 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
3 | onelss 4142 | . . . 4 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
5 | 4 | imp 122 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
6 | simpr 108 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
7 | elirr 4284 | . . . . 5 ⊢ ¬ 𝐵 ∈ 𝐵 | |
8 | 7 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ 𝐵) |
9 | 6, 8 | eldifd 2983 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (𝐴 ∖ 𝐵)) |
10 | eleq1 2141 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝐵 ∈ (𝐴 ∖ 𝐵))) | |
11 | 10 | spcegv 2686 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ (𝐴 ∖ 𝐵) → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))) |
12 | 6, 9, 11 | sylc 61 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) |
13 | phpm 6351 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵) | |
14 | 1, 5, 12, 13 | syl3anc 1169 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ≈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∃wex 1421 ∈ wcel 1433 ∖ cdif 2970 ⊆ wss 2973 class class class wbr 3785 Oncon0 4118 ωcom 4331 ≈ cen 6242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-er 6129 df-en 6245 df-dom 6246 |
This theorem is referenced by: (None) |
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