Theorem rexdif1enOLD 9083

Description: Obsolete version of rexdif1en 9082 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 8889	. 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 6394	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
4		f1ocnvdm 7226	. . . . . . 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 3442	. . . . . 6 ⊢ 𝑓 ∈ V
8		dif1enlemOLD 9081	. . . . . 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 4589	. . . . . . . 8 ⊢ (𝑥 = (◡𝑓‘𝑀) → {𝑥} = {(◡𝑓‘𝑀)})
11	10	difeq2d 4079	. . . . . . 7 ⊢ (𝑥 = (◡𝑓‘𝑀) → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {(◡𝑓‘𝑀)}))
12	11	breq1d 5105	. . . . . 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 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 2109  ∃wrex 3053  Vcvv 3438   ∖ cdif 3902  {csn 4579   class class class wbr 5095  ^◡ccnv 5622  suc csuc 6313  –1-1-onto→wf1o 6485  ‘cfv 6486  ωcom 7806   ≈ cen 8876

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7807  df-en 8880

This theorem is referenced by: (None)