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Mirrors > Home > MPE Home > Th. List > ssdomfi | Structured version Visualization version GIF version |
Description: A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 9027). (Contributed by BTernaryTau, 12-Nov-2024.) |
Ref | Expression |
---|---|
ssdomfi | ⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6882 | . . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
2 | f1of1 6843 | . . . . 5 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 → ( I ↾ 𝐴):𝐴–1-1→𝐴) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ( I ↾ 𝐴):𝐴–1-1→𝐴 |
4 | f1ss 6804 | . . . 4 ⊢ ((( I ↾ 𝐴):𝐴–1-1→𝐴 ∧ 𝐴 ⊆ 𝐵) → ( I ↾ 𝐴):𝐴–1-1→𝐵) | |
5 | 3, 4 | mpan 688 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴):𝐴–1-1→𝐵) |
6 | f1domfi 9215 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ ( I ↾ 𝐴):𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | |
7 | 5, 6 | sylan2 591 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵) → 𝐴 ≼ 𝐵) |
8 | 7 | ex 411 | 1 ⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3949 class class class wbr 5152 I cid 5579 ↾ cres 5684 –1-1→wf1 6550 –1-1-onto→wf1o 6552 ≼ cdom 8968 Fincfn 8970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-1o 8493 df-en 8971 df-dom 8972 df-fin 8974 |
This theorem is referenced by: php 9241 php2 9242 imadomfi 41505 |
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