Theorem rexdif1enOLD 9173

Description: Obsolete version of rexdif1en 9172 as of 5-Jan-2025. (Contributed by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rexdif1enOLD	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑀

Proof of Theorem rexdif1enOLD

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8969	. 2 ⊢ (𝐴 ≈ suc 𝑀 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀)
2		19.42v 1953	. . 3 ⊢ (∃𝑓(𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) ↔ (𝑀 ∈ ω ∧ ∃𝑓 𝑓:𝐴–1-1-onto→suc 𝑀))
3		sucidg 6435	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
4		f1ocnvdm 7278	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (◡𝑓‘𝑀) ∈ 𝐴)
5	4	ancoms 458	. . . . . 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 3463	. . . . . 6 ⊢ 𝑓 ∈ V
8		dif1enlemOLD 9171	. . . . . 6 ⊢ ((𝑓 ∈ V ∧ 𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)
9	7, 8	mp3an1 1450	. . . . 5 ⊢ ((𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀)
10		sneq 4611	. . . . . . . 8 ⊢ (𝑥 = (◡𝑓‘𝑀) → {𝑥} = {(◡𝑓‘𝑀)})
11	10	difeq2d 4101	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(◡𝑓‘𝑀)}))
12	11	breq1d 5129	. . . . . 6 ⊢ (𝑥 = (◡𝑓‘𝑀) → ((𝐴 ∖ {𝑥}) ≈ 𝑀 ↔ (𝐴 ∖ {(◡𝑓‘𝑀)}) ≈ 𝑀))
13	12	rspcev 3601	. . . . 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 1930	. . 3 ⊢ (∃𝑓(𝑀 ∈ ω ∧ 𝑓:𝐴–1-1-onto→suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
16	2, 15	sylbir 235	. 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)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459   ∖ cdif 3923  {csn 4601   class class class wbr 5119  ^◡ccnv 5653  suc csuc 6354  –1-1-onto→wf1o 6530  ‘cfv 6531  ωcom 7861   ≈ cen 8956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-en 8960

This theorem is referenced by: (None)