Theorem enp1i

Description: Proof induction for en2 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016) Generalize to all ordinals and avoid ax-pow , ax-un . (Revised by BTernaryTau, 6-Jan-2025)

	Ref	Expression
Hypotheses	enp1i.1	⊢ Ord 𝑀
	enp1i.2	⊢ 𝑁 = suc 𝑀
	enp1i.3	⊢ ( ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 → 𝜑 )
	enp1i.4	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
Assertion	enp1i	⊢ ( 𝐴 ≈ 𝑁 → ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		enp1i.1	⊢ Ord 𝑀
2		enp1i.2	⊢ 𝑁 = suc 𝑀
3		enp1i.3	⊢ ( ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 → 𝜑 )
4		enp1i.4	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
5	2	breq2i	⊢ ( 𝐴 ≈ 𝑁 ↔ 𝐴 ≈ suc 𝑀 )
6		encv	⊢ ( 𝐴 ≈ suc 𝑀 → ( 𝐴 ∈ V ∧ suc 𝑀 ∈ V ) )
7	6	simprd	⊢ ( 𝐴 ≈ suc 𝑀 → suc 𝑀 ∈ V )
8		sssucid	⊢ 𝑀 ⊆ suc 𝑀
9		ssexg	⊢ ( ( 𝑀 ⊆ suc 𝑀 ∧ suc 𝑀 ∈ V ) → 𝑀 ∈ V )
10	8 9	mpan	⊢ ( suc 𝑀 ∈ V → 𝑀 ∈ V )
11		elong	⊢ ( 𝑀 ∈ V → ( 𝑀 ∈ On ↔ Ord 𝑀 ) )
12	7 10 11	3syl	⊢ ( 𝐴 ≈ suc 𝑀 → ( 𝑀 ∈ On ↔ Ord 𝑀 ) )
13	1 12	mpbiri	⊢ ( 𝐴 ≈ suc 𝑀 → 𝑀 ∈ On )
14		rexdif1en	⊢ ( ( 𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ) → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 )
15	13 14	mpancom	⊢ ( 𝐴 ≈ suc 𝑀 → ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 )
16	3	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝐴 ∖ { 𝑥 } ) ≈ 𝑀 → ∃ 𝑥 ∈ 𝐴 𝜑 )
17		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
18	4	imp	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 )
19	18	eximi	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃ 𝑥 𝜓 )
20	17 19	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 𝜓 )
21	15 16 20	3syl	⊢ ( 𝐴 ≈ suc 𝑀 → ∃ 𝑥 𝜓 )
22	5 21	sylbi	⊢ ( 𝐴 ≈ 𝑁 → ∃ 𝑥 𝜓 )