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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 109 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ∈ ω) | |
2 | nnon 4624 | . . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
3 | onelss 4402 | . . . 4 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
5 | 4 | imp 124 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) |
6 | simpr 110 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
7 | elirr 4555 | . . . . 5 ⊢ ¬ 𝐵 ∈ 𝐵 | |
8 | 7 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ 𝐵) |
9 | 6, 8 | eldifd 3154 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (𝐴 ∖ 𝐵)) |
10 | eleq1 2252 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ 𝐵 ∈ (𝐴 ∖ 𝐵))) | |
11 | 10 | spcegv 2840 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐵 ∈ (𝐴 ∖ 𝐵) → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵))) |
12 | 6, 9, 11 | sylc 62 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) |
13 | phpm 6883 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴 ∧ ∃𝑥 𝑥 ∈ (𝐴 ∖ 𝐵)) → ¬ 𝐴 ≈ 𝐵) | |
14 | 1, 5, 12, 13 | syl3anc 1249 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ¬ 𝐴 ≈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∃wex 1503 ∈ wcel 2160 ∖ cdif 3141 ⊆ wss 3144 class class class wbr 4018 Oncon0 4378 ωcom 4604 ≈ cen 6756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-er 6553 df-en 6759 df-dom 6760 |
This theorem is referenced by: (None) |
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