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| Mirrors > Home > MPE Home > Th. List > rexdif1en | Structured version Visualization version GIF version | ||
| Description: If a set is equinumerous to a nonzero ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals and avoid ax-un 7756. (Revised by BTernaryTau, 5-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| rexdif1en | ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | encv 8994 | . . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ∈ V ∧ suc 𝑀 ∈ V)) | |
| 2 | 1 | simpld 494 | . . . 4 ⊢ (𝐴 ≈ suc 𝑀 → 𝐴 ∈ V) | 
| 3 | breng 8995 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ suc 𝑀 ∈ V) → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)) | |
| 4 | 1, 3 | syl 17 | . . . . . 6 ⊢ (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)) | 
| 5 | 4 | ibi 267 | . . . . 5 ⊢ (𝐴 ≈ suc 𝑀 → ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀) | 
| 6 | sucidg 6464 | . . . . . . . . . 10 ⊢ (𝑀 ∈ On → 𝑀 ∈ suc 𝑀) | |
| 7 | f1ocnvdm 7306 | . . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴) | |
| 8 | 7 | ancoms 458 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ suc 𝑀 ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴) | 
| 9 | 6, 8 | sylan 580 | . . . . . . . . 9 ⊢ ((𝑀 ∈ On ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴) | 
| 10 | 9 | adantll 714 | . . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴) | 
| 11 | vex 3483 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
| 12 | dif1enlem 9197 | . . . . . . . . 9 ⊢ (((𝑓 ∈ V ∧ 𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) | |
| 13 | 11, 12 | mp3anl1 1456 | . . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) | 
| 14 | sneq 4635 | . . . . . . . . . . 11 ⊢ (𝑥 = (◡𝑓‘𝑀) → {𝑥} = {(◡𝑓‘𝑀)}) | |
| 15 | 14 | difeq2d 4125 | . . . . . . . . . 10 ⊢ (𝑥 = (◡𝑓‘𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(◡𝑓‘𝑀)})) | 
| 16 | 15 | breq1d 5152 | . . . . . . . . 9 ⊢ (𝑥 = (◡𝑓‘𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)) | 
| 17 | 16 | rspcev 3621 | . . . . . . . 8 ⊢ (((◡𝑓‘𝑀) ∈ 𝐴 ∧ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) | 
| 18 | 10, 13, 17 | syl2anc 584 | . . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝑀 ∈ On) ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) | 
| 19 | 18 | ex 412 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝑓:𝐴–1-1-onto→suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)) | 
| 20 | 19 | exlimdv 1932 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)) | 
| 21 | 5, 20 | syl5 34 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)) | 
| 22 | 2, 21 | sylan 580 | . . 3 ⊢ ((𝐴 ≈ suc 𝑀 ∧ 𝑀 ∈ On) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)) | 
| 23 | 22 | ancoms 458 | . 2 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑀 → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)) | 
| 24 | 23 | syldbl2 841 | 1 ⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 ∖ cdif 3947 {csn 4625 class class class wbr 5142 ◡ccnv 5683 Oncon0 6383 suc csuc 6385 –1-1-onto→wf1o 6559 ‘cfv 6560 ≈ cen 8983 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-en 8987 | 
| This theorem is referenced by: findcard2 9205 enp1i 9314 | 
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