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Mirrors > Home > MPE Home > Th. List > php2 | Structured version Visualization version GIF version |
Description: Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.) Avoid ax-pow 5386. (Revised by BTernaryTau, 20-Nov-2024.) |
Ref | Expression |
---|---|
php2 | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnfi 9229 | . . 3 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin) | |
2 | pssss 4115 | . . 3 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴) | |
3 | ssdomfi 9258 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
4 | 3 | imp 406 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ≼ 𝐴) |
5 | 1, 2, 4 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≼ 𝐴) |
6 | php 9269 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵) | |
7 | ensymfib 9246 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) | |
8 | 7 | biimprd 248 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵)) |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵)) |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → (𝐵 ≈ 𝐴 → 𝐴 ≈ 𝐵)) |
11 | 6, 10 | mtod 198 | . 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐵 ≈ 𝐴) |
12 | brsdom 9031 | . 2 ⊢ (𝐵 ≺ 𝐴 ↔ (𝐵 ≼ 𝐴 ∧ ¬ 𝐵 ≈ 𝐴)) | |
13 | 5, 11, 12 | sylanbrc 582 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2103 ⊆ wss 3970 ⊊ wpss 3971 class class class wbr 5169 ωcom 7899 ≈ cen 8996 ≼ cdom 8997 ≺ csdm 8998 Fincfn 8999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-om 7900 df-1o 8518 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 |
This theorem is referenced by: php3 9271 php4 9272 nndomog 9275 nndomogOLD 9285 |
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