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Mirrors > Home > MPE Home > Th. List > phplem3 | Structured version Visualization version GIF version |
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (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 6252 | . 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) | |
2 | phplem2.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | phplem2.2 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | phplem2 8691 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
5 | 2 | enref 8536 | . . . 4 ⊢ 𝐴 ≈ 𝐴 |
6 | nnord 7582 | . . . . . 6 ⊢ (𝐴 ∈ ω → Ord 𝐴) | |
7 | orddif 6279 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴})) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴})) |
9 | sneq 4571 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
10 | 9 | difeq2d 4099 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
11 | 10 | eqcoms 2829 | . . . . 5 ⊢ (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵})) |
12 | 8, 11 | sylan9eq 2876 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵})) |
13 | 5, 12 | breqtrid 5096 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
14 | 4, 13 | jaodan 954 | . 2 ⊢ ((𝐴 ∈ ω ∧ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴)) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
15 | 1, 14 | sylan2 594 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∖ cdif 3933 {csn 4561 class class class wbr 5059 Ord word 6185 suc csuc 6188 ωcom 7574 ≈ cen 8500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-om 7575 df-en 8504 |
This theorem is referenced by: phplem4 8693 php 8695 |
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