![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > phplem1 | GIF version |
Description: Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.) |
Ref | Expression |
---|---|
phplem1 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnord 4626 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | nordeq 4558 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) | |
3 | disjsn2 3670 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) |
5 | 1, 4 | sylan 283 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) |
6 | undif4 3500 | . . 3 ⊢ (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (({𝐴} ∪ 𝐴) ∖ {𝐵})) | |
7 | df-suc 4386 | . . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
8 | 7 | equncomi 3296 | . . . 4 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) |
9 | 8 | difeq1i 3264 | . . 3 ⊢ (suc 𝐴 ∖ {𝐵}) = (({𝐴} ∪ 𝐴) ∖ {𝐵}) |
10 | 6, 9 | eqtr4di 2240 | . 2 ⊢ (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵})) |
11 | 5, 10 | syl 14 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∖ cdif 3141 ∪ cun 3142 ∩ cin 3143 ∅c0 3437 {csn 3607 Ord word 4377 suc csuc 4380 ωcom 4604 |
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-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 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-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-tr 4117 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 |
This theorem is referenced by: phplem2 6871 |
Copyright terms: Public domain | W3C validator |