Theorem iotassuni 4355

Description: The ℩ class is a subset of the union of all elements satisfying φ. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
iotassuni	⊢ (℩xφ) ⊆ ∪{x ∣ φ}

Proof of Theorem iotassuni

Step	Hyp	Ref	Expression
1		iotauni 4351	. . 3 ⊢ (∃!xφ → (℩xφ) = ∪{x ∣ φ})
2		eqimss 3323	. . 3 ⊢ ((℩xφ) = ∪{x ∣ φ} → (℩xφ) ⊆ ∪{x ∣ φ})
3	1, 2	syl 15	. 2 ⊢ (∃!xφ → (℩xφ) ⊆ ∪{x ∣ φ})
4		0ss 3579	. . 3 ⊢ ∅ ⊆ ∪{x ∣ φ}
5		iotanul 4354	. . . 4 ⊢ (¬ ∃!xφ → (℩xφ) = ∅)
6	5	sseq1d 3298	. . 3 ⊢ (¬ ∃!xφ → ((℩xφ) ⊆ ∪{x ∣ φ} ↔ ∅ ⊆ ∪{x ∣ φ}))
7	4, 6	mpbiri 224	. 2 ⊢ (¬ ∃!xφ → (℩xφ) ⊆ ∪{x ∣ φ})
8	3, 7	pm2.61i 156	1 ⊢ (℩xφ) ⊆ ∪{x ∣ φ}

Syntax hints:  ¬ wn 3   = wceq 1642  ∃!weu 2204  {cab 2339   ⊆ wss 3257  ∅c0 3550  ∪cuni 3891  ℩cio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339

This theorem is referenced by: (None)