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Mirrors > Home > ILE Home > Th. List > phplem3 | GIF version |
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6342. (Contributed by NM, 26-May-1998.) |
Ref | Expression |
---|---|
phplem2.1 | ⊢ 𝐴 ∈ V |
phplem2.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
phplem3 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuci 4158 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | phplem2.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | phplem2.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | phplem2 6339 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
5 | 2 | enref 6268 | . . . 4 ⊢ 𝐴 ≈ 𝐴 |
6 | nnord 4352 | . . . . . 6 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
7 | orddif 4290 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | |
8 | 6, 7 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | sneq 3409 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
10 | 9 | difeq2d 3090 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
11 | 10 | eqcoms 2084 | . . . . 5 ⊢ (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
12 | 8, 11 | sylan9eq 2133 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵})) |
13 | 5, 12 | syl5breq 3820 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
14 | 4, 13 | jaodan 743 | . 2 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
15 | 1, 14 | sylan2 280 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∨ wo 661 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ∖ cdif 2970 {csn 3398 class class class wbr 3785 Ord word 4117 suc csuc 4120 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-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-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-en 6245 |
This theorem is referenced by: phplem4 6341 phplem3g 6342 |
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