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Mirrors > Home > MPE Home > Th. List > rexdif1enOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rexdif1en 9098 as of 5-Jan-2025. (Contributed by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rexdif1enOLD | ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 8889 | . 2 ⊢ (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀) | |
2 | 19.42v 1957 | . . 3 ⊢ (∃𝑓(𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) ↔ (𝑀 ∈ ω ∧ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)) | |
3 | sucidg 6396 | . . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀) | |
4 | f1ocnvdm 7227 | . . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴) | |
5 | 4 | ancoms 459 | . . . . . 6 ⊢ ((𝑀 ∈ suc 𝑀 ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴) |
6 | 3, 5 | sylan 580 | . . . . 5 ⊢ ((𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴) |
7 | vex 3447 | . . . . . 6 ⊢ 𝑓 ∈ V | |
8 | dif1enlemOLD 9097 | . . . . . 6 ⊢ ((𝑓 ∈ V ∧ 𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) | |
9 | 7, 8 | mp3an1 1448 | . . . . 5 ⊢ ((𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) |
10 | sneq 4594 | . . . . . . . 8 ⊢ (𝑥 = (◡𝑓‘𝑀) → {𝑥} = {(◡𝑓‘𝑀)}) | |
11 | 10 | difeq2d 4080 | . . . . . . 7 ⊢ (𝑥 = (◡𝑓‘𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(◡𝑓‘𝑀)})) |
12 | 11 | breq1d 5113 | . . . . . 6 ⊢ (𝑥 = (◡𝑓‘𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)) |
13 | 12 | rspcev 3579 | . . . . 5 ⊢ (((◡𝑓‘𝑀) ∈ 𝐴 ∧ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) |
14 | 6, 9, 13 | syl2anc 584 | . . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) |
15 | 14 | exlimiv 1933 | . . 3 ⊢ (∃𝑓(𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) |
16 | 2, 15 | sylbir 234 | . 2 ⊢ ((𝑀 ∈ ω ∧ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) |
17 | 1, 16 | sylan2b 594 | 1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃wrex 3071 Vcvv 3443 ∖ cdif 3905 {csn 4584 class class class wbr 5103 ◡ccnv 5630 suc csuc 6317 –1-1-onto→wf1o 6492 ‘cfv 6493 ωcom 7798 ≈ cen 8876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-om 7799 df-en 8880 |
This theorem is referenced by: (None) |
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