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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 4583 | . . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
2 | nordeq 4515 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵) | |
3 | disjsn2 3633 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) |
5 | 1, 4 | sylan 281 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ({𝐴} ∩ {𝐵}) = ∅) |
6 | undif4 3466 | . . 3 ⊢ (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (({𝐴} ∪ 𝐴) ∖ {𝐵})) | |
7 | df-suc 4343 | . . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) | |
8 | 7 | equncomi 3263 | . . . 4 ⊢ suc 𝐴 = ({𝐴} ∪ 𝐴) |
9 | 8 | difeq1i 3231 | . . 3 ⊢ (suc 𝐴 ∖ {𝐵}) = (({𝐴} ∪ 𝐴) ∖ {𝐵}) |
10 | 6, 9 | eqtr4di 2215 | . 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 103 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ∖ cdif 3108 ∪ cun 3109 ∩ cin 3110 ∅c0 3404 {csn 3570 Ord word 4334 suc csuc 4337 ωcom 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 |
This theorem is referenced by: phplem2 6810 |
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